Multi-electron AtomsIntroductionConsider the Helium atomH e2 +e-e-r1r2r1 2Hamiltonian includes1) Kinetic energy of electron 1 and electron 2- Relative coordinates are used so that kinetic energy of nucleus is essentially center-of-mass motion of atom.2) Coulomb attraction between electron 1 and nucleus3) Coulomb attraction between electron 2 and nucleus4) Coulomb repulsion between electron 1 and electron 22 2 2 2 21 21 2 0 1 0 2 0 12ˆ ˆp p 1 Ze 1 Ze 1 e2 2 4 r 4 r 4 r        Hremember that 2 2 2ˆp  hAs with the other multivariable differential equations that we have encountered, we would like to attempt a separation of the r1 and r2 variables, so that      1 2 1 2r , r r r  - The Coulomb repulsion term makes an analytic separation of variables impossible.- Separate orbitals for the electrons can be found approximately using the iterative, self-consistent field method called the Hartree-Fock method.Hartree-Fock Method11. Start with hydrogen wave functions for each electron. Assume a separation of variables.2. Create an effective Coulomb repulsion potential energy function for electron 1 due to the interaction with electron 2.     *1 2 2 2121V r r r drr       *2 1 1 1121V r r r drr  Note: At this point, the potential energy is only dependent on r1 or r2, not r1 andr2 together. But the potential energy we have is only approximate.3. The Hamiltonian becomes.1 2 H H H 2 2 211 11 0 11 ZeV r2 4 r    hH 2 2 222 22 0 21 ZeV r2 4 r    hH4. Now that the Schrödinger equation can be separated into two single-particle Schrödinger equations, solve them to find new eigenfunctions.   1 1 1 1r E r   H   2 2 2 2r E r   HNote: The prime notation is used since, after solving the Schrödinger equation, the wavefunctions are not the hydrogen wavefunctions but modified wavefunctions.5. Go back to step 2 to create a new (and better) effective Coulomb repulsion potential energy function. Repeat steps 3 and 4.6. Repeat steps 2, 3, 4, 5 until the latest energy eigenvalues equal to (or more precisely, close enough to) the previous energy eigenvalues.Our method has converged on a single, unique solution for the eigenvalues and the eigenfunctions for our problem. In addition, our solution is a product of single electron wavefunctions. Such wavefunctions are very convenient for describing the electronic structure of atoms. However, it is worth emphasizing that such a product is only an approximation.Spinors and Spin Orbitals2Thus far, we have not considered the spin of the electron in the structure of atoms. From the General Chemistry course, we learned that the periodic table could not have the structure that it does without including the fact that the electron has spin.Spin Quantum NumberClassically a system with a charge moving in a circle (i.e., a charge with angular momentum) creates what is called a magnetic moment. This magnetic moment will twist in the direction of a magnetic field gradient.In 1922, Otto Stern and Walter Gerlach, put created a beam of silver atoms and put the beam into a magnetic field gradient. The beam of atoms was split in two with some atoms being deflected in one direction and the other atoms being deflected in the oppositedirection. This deflection must have been caused by a magnetic moment. But from where does the magnetic moment in a silver atom come? Recall that the ground electronic configuration of the silver atom is [Kr] 5s14d10. The closed [Kr] and 4d10 shell should not be able to create any magnetic moment in the atom. But then again neither should the 5s electron since it has zero angular momentum.B o x o fA g v a p o rN o r t h M a g n e t i c P o l eS o u t h M a g n e t i c P o l eS p i n u pS p i n d o w nP h o t o g r a p h i cP l a t eIn 1925, Samuel Goudsmit and George Uhlenbeck, as graduate students presumed that the magnetic moment in the silver atom came from intrinsic angular momentum in the electron. Such intrinsic angular momentum could be compared to the electron spinning on its axis. They proposed that the electron has only two possible spins along with two possible values of a new quantum number, ms.Ms = + ½ “spin up” sometimes labeled Ms = - ½ “spin down” sometimes labeled Problem: Schrödinger equation did not yield spin quantum number.3The Dirac equation (the relativistic four-dimensional Schrödinger equation) does yield the spin quantum number. However, we would prefer to keep things simple. Therefore, we will treat spin in an ad hoc manner.SpinorsTo introduce spin as a mathematical function to the wavefunction, we multiply the wavefunction we get from the Schrödinger equation by a two-dimensional vector called aspinor. 1s1r, ,0      spin up 1s0r, ,1      spin downThese wavefunctions that include their spin as part of their description are called spin orbitals.Note that 1s and 1s are orthogonal.( ) ( )21s 1s 1s0d r, , d 1 0 01a b��y y t = y q f t =����� �With the proper operators, which would be in matrix form, we can calculate any observable as we had done before.Though spinors are used to describe spin mathematically, we shall not use them. We label spin orbitals as  or  orbitals.Indistinguishability and the Pauli Exclusion Principle4IndistinguishabilityConsider the two electrons in a Helium atom.Which electron is electron 1 and which is electron 2?- Take picture and label electrons- Uh, oh! Measuring position disturbs the system, now where is electron?- Since we can’t measure position and momentum together with infinite precision, we can’t tell electrons apart?Thus, the electrons are indistinguishable.The indistinguishability of the electrons is not a problem of our technology; it is a fundamental property of microscopic particles.Pauli Exclusion PrincipleWeaker statement: No two electrons can have the same set of quantum numbers.Stronger statement: Indistinguishable fermions must have total antisymmetric wavefunction.- fermion – particle with half-integral spin, i.e., e-, p+, 13C.- boson – particle with integral spin, photon, 2D, 14N, H atom.5H e2