Restricted and Unrestricted Hartree-Fock method Sudarshan Dhungana Phys790 Seminar (Feb15,2007) OVERVIEW Introduction Approximations Hartree and Hartree-Fock methods Derivation of HF equations Basis functions Solution of RHF equation Results Advantages and Disadvantages References Introduction The HF method is widely used (mostly in the Quantum physics,quantum chemistry community) because By itself it can provide sufficiently accurate results, generally In atoms and molecules If we have the HF solution,the accuracy can systematically be improved by applying various techniques(configuration interaction The HF method is based on Variational principle Introduction HF calculations with identical spatial orbitals for electrons with spins up/down are called restricted Hartree-Fock method If different orbitals are used,we have unrestricted Hartree-Fock method Closed and open shell systems If the number of electrons is even and S=0,we have a closed shell (fig a) If the number of electrons is odd,we have an open-shell system (fig.b) In general,if the numbers of electrons with spins up and down are different,we have an open-shell system Approximations Born-Oppenheimer approach:the nuclei are fixed The relativistic effects are completely neglected The real Hamiltonian is essentilly replaced with a single electron Hamiltonian Approximations The total WF can be represented as a single slater determinant but from the spin- orbitals Difference between Hartree and Hartree Fock method Hartree-method: Multi electron wave function is approximated by a product of single electron wave functions Hartree-Fock method: Multi-electron wave function is approximated by a antisymmetrized sum of products of single electron wave function Derivation of the Hartree-Fock equation The Hamiltonian reads: where Derivation of Hartree Fock equation We write the total electron wave function as a Slater determinant built from orthonormal spin-orbitals, Note that if spin orbitals are normalized, is also normalized In what follows,the integrals means the integration over both spatial and sum over spin coordinates Now we calculate the total energy as, Derivation of Hartree Fock equation First part of total energy: The second part of total energy: Where Derivation of Hartree Fock equation Total energy: We now define the Coulomb operator, And exchange operator, Derivation of Hartree Fock equation In terms of these operators,we can write the energy as Now we determine the minimum of the functional as a function of the spin orbitals k provided that they meet the orthonormality condition: This implies that we have a minimisation problem with constraints,so we make use of the Lagrange multipliers - =0 Derivation of Hartree Fock equation Where, Using the symmetry, Introducing a new Hermitian Operator (Fock operator) Derivation of Hartree Fock equation Because of the symmetry of the constraint equation,we must have Now : And from the expression, - =0 we arrive at: Derivation of Hartree Fock equation Since is small but arbitrary, which with leads to The Lagrange parameter in the above equation can not be chosen freely: they must be such that the solutions form an orthonormal set.An obvious solution of the above equation is found by taking the as the eigen vectors of the Fock operators with eigen values and Derivation of Hartree Fock equation We have Schrdinger type equation Can be called orbitals Can be called orbital energies The total energy The Coulomb and exchange contributions must be subtracted off from the sum of Fock levels (to avoid counting the two electron integrals twice) Derivation of Hartree Fock equation Some remarks The sum here runs over all occupied levels Eq. Is non linear, it must be solved by self- consistency procedure Solving eq. Yields an infinite spectrum: to find the ground state,we take the lowest N eigen states of the spectrum as the electron spin-orbitals. these orbitals are used to build a new Fock operator,then the procedure is repeated over and over until the convergence is reached. Solutions of Hartree fock equation In order to solve the eigen value problem for the Fock operator, we have to expand the spin-orbitals in a basis set, that is we have to represent the orbitals as linear combinations of a finite no of basis states Plane waves perfect for crystals (periodic system) Slater functions Solutions for electrons in isolated atoms/small molecules Gaussian functions Solutions of Hartree fock equation The general form of Fock operator is, With It is possible to eliminate the spin degrees of freedom by summing over them and find an operator acting only on spatial orbitals Solutions of Hartree fock equation The coulomb and exchange operators,written in terms of orbital parts only read The Fock operator now becomes, Now for a given basis we obtain the following relation Solutions of Hartree fock equation The Fock matrix is now given by Where, Where k labels the orbitals and p,q,r label the basis functions We introduce the density matrix, which (for RHF) is defined as Solutions of Hartree Fock equation The physical meaning of the Fock matrx is transparent :motion of an electron in the field of the nuclei and other electrons.Using density matrix Fock matrix is written as And the total energy is Unrestricted Hartree Fock Method The spin up orbitals are represented by and spin down orbitals are represented by With this vectors we reformulate HF equation with Solutions of Hartree fock equation Application of restricted HF method (Closed shell atom) For beryllium Application of restricted HF method (open shell atom) For Chlorine Advantages and Disadvantages References Hartree Fock Program Dr.Peter Winkler Australian Journal of physics Dr.J.Mitroy Computational physics Dr.J.M.Thijssen Inroduction to electronic structure simulationsInroduction to electronic structure simulations Dr.Arkady Krasheninnikov