1. Lecture 3. Many-Electron Atom. Pt.1 Electron Correlation, Indistinguishableness, Antisymmetry & Slater Determinant References Ratner Ch. 7.1-7.2, 8.1-8.5, Engel Ch. 10.1-10.3, Pilar Ch.7 Modern Quantum Chemistry, Ostlund & Szabo, Ch. 2.1-2.2 Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7 Quantum Chemistry, McQuarrie, Ch. 7-8 Computational Chemistry, Lewars (2003), Ch.4 A Brief Review of Elementary Quantum Chemistry http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html Slater, J.; Verma, H.C. (1929) Phys. Rev. 34, 1293-1295.

2. Electron-electron repulsion Indistinguishability Helium Atom First (1 nucleus + 2 electrons) We cannot solve this Schrödinger equation analytically. (Two electrons are not separable nor independent any more.)  A series of approximations will be introduced. 1. Electron-electron repulsion (correlation) The r 12 term removes the spherical symmetry in He. ~H atom electron at r 1 ~H atom electron at r 2 newly introduced : Correlated, coupled

3. Approximation #1. Orbital Approximation or Hartree Approximation or Single-particle approach or One-body approach To first approximation, electrons are treated independently. An N-electron wave function is approximated by a product of N one-electron wave functions (orbitals) (a so-called Hartree product). This does not mean that electrons do not sense each other. (We’ll see later.) 1-electron wave function ~ H atom orbital Many-electron (many-body) wave function

4. Electron has an “intrinsic spin” angular momentum, which has nothing to do with the orbital angular momentum in an atom. Electron spin & Spin angular momentum The Stern-Gerlach experiment shows two beams. Multiplicity = 2 (doublet) l = 0, 1, 2, …; multiplicity (= the number of allowed m l values) = 1, 3, 5, …

5. Spin operator, Eigenfunctions, and Eigenvalues Spin (angular momentum) operator, s 2 and s z : - just like orbital angular momentum operator, L 2 and L z Two eigenstates only { ,  } or { ,  } – an orthonormal set: - eigenfunction  with eigenvalue (s, m s ) = (½, ½) - eigenfunction  with eigenvalue (s, m s ) = (½, -½) - We don’t know (don’t care) the form of the eigenfunction  and .  (  )  (  ) ½ -½  3/2

6. space spin How is spin integrated into wave function?  |1s>  Just a product of spatial orbital  spin orbital, because the non-relativistic Hamiltonian operator does not include spin. (Space & spin variables are separated, and [H,s 2 ] = [H,s z ] = [s 2,s z ] = 0) Orthogonal to each other (integration now over r, , , and  ) 1s e.g. new degree of freedom (4 th quantum number) with only two values (1/2, -1/2) - Wolfgang Pauli (1924)

7. 1885 – Johann Balmer – Line spectrum of hydrogen 1886 – Heinrich Hertz – Photoelectric effect experiment 1897 – J. J. Thomson – Discovery of electrons from cathode rays experiment 1900 – Max Planck – Quantum theory of blackbody radiation 1905 – Albert Einstein– Quantum theory of photoelectric effect 1910 – Ernest Rutherford – Scattering experiment with  -particles 1913 – Niels Bohr – Quantum theory of hydrogen spectra 1923 – Arthur Compton – Scattering experiment of photons off electrons 1924 – Wolfgang Pauli – Exclusion principle – Ch. 10 1924 – Louis de Broglie – Matter waves 1925 – Davisson and Germer – Diffraction experiment on wave properties of electrons 1926 – Erwin Schrodinger – Wave equation – Ch. 2 1927 – Werner Heisenberg – Uncertainty principle – Ch. 6 1927 – Max Born – Interpretation of wave function – Ch. 3 History of Quantum Mechanics particle wave & spin

8. 1922 – Otto Stern & Walter Gerlach – The existence of spin angular momentum is inferred from their experiment, in which particles (Ag atoms) are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone. 1924 – Wolfgang Pauli – proposed a new quantum degree of freedom (or quantum number) with two possible values and formulated the Pauli exclusion principle. 1925 – Ralph Kronig, George Uhlenbeck & Samuel Goudsmit – identified Pauli's new degree of freedom as electron spin and suggested a physical interpretation of particles spinning around their own axis. 1926 – Enrico Fermi & Paul Dirac – formulated (independently) the Fermi-Dirac statistics, which describes distribution of many identical particles obeying the Pauli exclusion principle (fermions with half-integer spins – contrary to bosons satisfying the Bose-Einstein statistics) 1926 – Erwin Schrödinger – formulated his non-relativistic Schrödinger equation, but it incorrectly predicted the magnetic moment of H to be zero in its ground state. 1927 – T.E. Phipps & J.B. Taylor – reproduced the effect using H atoms in the ground state, thereby eliminating any doubts that may have been caused by the use of Ag atoms. 1927 – Wolfgang Pauli – worked out on mathematical formulation of spin (2  2 matrices). 1928 – Paul Dirac – showed that spin comes naturally from his relativistic Dirac equation. A Little History of Spin in Quantum Mechanics

9. 2. Electrons (in a He atom) are indistinguishable. Two possibilities in wave function  Probability doesn’t change. 1s e.g. asymmetric not good! ok

10. Electrons (s = ½) are fermion (s = half-integer).  antisymmetric wavefunction Antisymmetry of electrons (or other fermions) Quantum postulate 6 (Pauli Principle; 1924-1925): Wave functions describing a many-electron system should -change sign (be antisymmetric) under the exchange of any two electrons. -be an eigenfunction of the exchange operator P 12 with the eigenvalue of -1. not ok! ok P 12  exchange operator [H, P 12 ] = 0

11. Ground state of He (the singlet state) Slate determinants provide a convenient way to antisymmetrize many-electron wave functions built with the Hartree approximation. 1s Slater determinant (1929) notation  |1s>  1s S 2  (1,2) = (s 1 + s 2 ) 2  (1,2) = 0, S z  (1,2) = (s z1 + s z2 )  (1,2) = 0  1s 2

12. Excited state of He

13. Slater determinant and Pauli exclusion principle A determinant changes sign when two rows (or columns) are exchanged.  Exchanging two electrons leads to a change in sign of the wave function. A determinant with two identical rows (or columns) is equal to zero.  No two electrons can occupy the same state. “Pauli’s exclusion principle” “antisymmetric” = 0 4 quantum numbers (space and spin)  We cannot put more than two electrons in one space orbital (nlm l ).

14. N-electron wave function: Slater determinant N-electron wave function is approximated by a product of N one-electron wave functions (hartree product). It should be antisymmetrized. but not antisymmetric!

15. Ground state of Lithium

16. Total angular momentum of many-electron atom Add l i (or s i ) vectors to form L (or S) vector. LS coupling (contrary to jj coupling) for non-relativistic, no-spin-orbit-coupling cases

17. Ground state of He (the singlet state) 1s Slater determinant notation  |1s>  1s S 2  (1,2) = (s 1 + s 2 ) 2  (1,2) = 0, S z  (1,2) = (s z1 + s z2 )  (1,2) = 0  1s 2 Total spin quantum number S = s 1 + s 2 = ½ - ½ = 0, M s = 0 (singlet)

18. Excited state of He (singlet and triplet states)

19. S = s 1 + s 2 = ½ + ½ = 1, M s = 1, 0, -1 (triplet) S = s 1 + s 2 = ½ - ½ = 0, M s = 0 (singlet) (S,M s ) (0,0) (1,1) (1,-1) (1,0)    4 = -4 =-4 = 2 =2 = 3 =3 = + Excited state of He (singlet and triplet states)