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Lecture 5. Many-Electron Atoms. Pt.3 Hartree-Fock Self-Consistent-Field Method References • • • • Ratner Ch. 9, Engel Ch. 10.5, Pilar Ch. 10 Modern Quantum Chemistry, Ostlund & Szabo (1982) Ch. 2-3.4.5 Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7 Computational Chemistry, Lewars (2003), Ch.4 • A Brief Review of Elementary Quantum Chemistry http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html Hartree (single-particle) self-consistent-field method based on Hartree products (D. R. Hartree, 1928) Proc. Cambridge Phil. Soc. 24, 89 • 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) Nobel lecture (Walter Kohn; 1998) Electronic structure of matter Constrained minimization with the Hartree product Hartree-Fock Self-Consistent-Field Method based on Slater determinants (Hartree+Pauli) (J. C. Slater; V. Fock, 1930) Z. Physik, 61, 126; Phys. Rev. 35, 210 Restrict the search for the minimum E[] to a subset of , a Slater determinant. • To build many-electron wave functions, assume that electrons are uncorrelated. (Hartree products of one-electron orbitals) • To build many-electron wave functions, use Slater determinants, which is all antisymmetric products of N spin orbitals, to satisfy the Pauli principle. • Use the variational principle to find the best Slater determinant (which yields the lowest energy) by varying the spatial orbitals {i}. Beyond Hartree: the ground state of He (singlet) notation |1s> Slater determinant 1s Total spin quantum number S = Ms = 0 (singlet) S2 (1,2) = (s1 + s2)2 (1,2) = 0, Sz (1,2) = (sz1 + sz2) (1,2) = 0 1s 1s2 Energy of the Slater determinant of the He atom: the “singlet” ground state spatial-symmetric spin-antisymmetric no spin in the Hamiltonian Coulombic repulsion between two charge distributions |1s(1)|2 and |1s(2)|2 Coulomb integral <1s|h|1s> =<1s|T+VNe|1s>=Tss+Vs Coulombic repulsion between two charge distributions |1s(1)|2 and |1s(2)|2 Excited state of He (singlet and triplet states) antisymmetric spatial-symmetric spinsymmetric spatial-antisymmetric spatial-symmetric spatial-antisymmetric Energy of the Slater determinant of the He atom: a “triplet” first excited state triplet singlet ? (quiz) Coulomb integral > 0 where Exchange integral (>0) includes in it wave function (final solution)! Energy of the Slater determinant of the He atom: a “triplet” first excited state triplet singlet Coulomb integral > 0 where Exchange integral (>0) includes in it wave function (final solution)! Two-electron interactions (Vee) • Coulomb integral Jij (local) > 0, i.e., a destabilization 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 Each term includes the wave function (the final solution) in it! Hartree-Fock Self-Consistent-Field Method based on Slater determinants (Hartree+Pauli) (J. C. Slater & V. Fock, 1930) • Each has variational parameters (to be changed to minimize E) including the effective nuclear charge (instead of the formal nuclear charge Z) or its combination for lower E • Variational condition = ij • Variation with respect to the one-electron orbitals {i}, which are orthonormal Constrained (due to the orthonormality of {i}) minimization of EHF[SD] leads to the HF equation. vergil.chemistry.gatech.edu/notes/hf-intro/node7.html ; Pilar Ch.10.1, Ostlund/Szabo Ch.1.3 Constrained minimization with the Slater determinant After constrained minimization with the Slater determinant Hartree-Fock equation (one-electron equation) Fock operator: “effective” one-electron operator Two-electron repulsion operator (1/rij) is replaced by one-electron operator VHF(i), which takes it into account in an “average” way & Two-electron repulsion cannot be separated exactly into one-electron terms. By imposing the separability, the orbital approximation inevitably involves an incorrect treatment of the way in which the electrons interact with each other. Hartree-Fock Self-Consistent Field (HF-SCF) Method Problem: •Fock operator (V) depends on the solution. •The answer (solution) must be known in order to solve the problem! •HF is not a regular eigenvalue problem that can be solved in a closed form. Solution (iterative approach): 1.Start with a guessed set of orbitals. 2.Solve the Hartree-Fock equation. 3.Use the resulting new set of orbitals in the next iteration and so on 4.Until the input and output orbitals differ by less than a preset threshold (i.e. converged to a self-consistent field). Hartree-Fock equation (One-electron equation) spherically symmetric Veff includes & - Two-electron repulsion operator (1/rij) is replaced by one-electron operator VHF(i), which takes it into account in an “average” way. - Any one electron sees only the spatially averaged position of all other electrons. - VHF(i) is spherically symmetric. - (Instantaneous) electron correlation is ignored. - Spherical harmonics (s, p, d, …) are valid angular-part eigenfunction (as for H-like atoms). - Radial-part eigenfunction of H-like atoms are not valid any more. optimized Electron Correlation Ref) F. Jensen, Introduction to Computational Chemistry, 2nd ed., Ch. 4 • A single Slater determinant never corresponds to the exact wave function. 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), MCSCF, CAFSCF, … Solution of HF-SCF equation gives Solution of HF-SCF equation: Effective nuclear charge (Z- is a measure of shielding.) Aufbau (Building-up) principle