# MANY ELECTRON ATOMS Chapter 15

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1 MANY ELECTRON ATOMS Chapter 15 Electron-Electron Repulsions ( ) The hydrogen atom Schrödinger equation is exactly solvable yielding the wavefunctions and orbitals of chemistry. Howev er, the Schrödinger equation cannot be analytically solved when there is more than one electron. Why? Consider the general Schrödinger equation for N electrons: 2 = 2 x y z 2 Were it not for the last term the above Hamiltonian would just be a sum of hydrogen atom hamiltonians for each electron. This last term represents electron-electron repulsion. It may be better understood if we examine the simplest multi-electron Hamiltonian, that for helium coulombic attraction between nucleus and electron kinetic energy of electron coulombic repulsion between electrons Since we have two or more electrons,the Schrödingeröequation cannot be solved exactly (a fundamental challenge for quantum chemistry))because of the last electron repulsion term - the motion of the electrons are correlated. If they did not interact the energy would just be a sum of each electrons's energy and the wavefunction a simple product. The Hartree-Fock Self-Consistent Field Method (15.8) How do we approach this problem?? Well,,try first keeping the simple product form for the wavefunction and use this in the Schrödinger equation: An N-electron wavefunction is approximated by a product of N one-electron wavefunctions (orbitals)) called a Hartree product. This does not mean that the electrons do not sense each other. This approach is called the Hartree Approximation or the Orbital Approximation. The Hartree-Fock self-consistent field method combined with the variational method is used to approximate eigenfunctions and eigenvalues of total energy for the many-electron atom. The variational theorem states that the energy of any approximation is always greater than or equal to the true energy of the system ((what you would obtain if you could solve the Schrödinger equation) for the N-electron atom.

2 -2- Basics of the approach The wavefunctions contain all the information possible. Where do we get intuition from them? First look at the hydrogen atom wavefunctions 1s R 10 (r) = 2 Z a 0 2s R 20(r) = 1 8 2p R 21(r) = s R 30(r) = 3p R 31(r) = 3d R 32(r) = e Zr/a 0 Z a 0 2 Zr Z Zr a 0 a 0 a 0 e Zr/2a 0 e Zr/2a 0 2 Z 18Zr Z 2 r a 0 a 0 4 Z 81 6 a 0 6Zr a 0 Z 2 r 2 a Z Z a 0 2 r 2 a 2 0 e Zr/3a 0 a 2 0 e Zr/3a 0 e Zr/3a 0 Notice how there are some general characteristics:: a factor of Z/a 00 and a decreasing exponential with the same factor, so try this form. ζ where zeta, ζ, is the effective nuclear charge. a 0 e ζr/a 0

3 -3- "...I have tried to base a system on the magnitudes of the atomic weights of the elements. My first attempt in this respect was the following: I chose the smallest atomic weights and arranged them according to the sizes of their atomic weights. This showed that there existed a periodicity in the properties of these simple substances and that even according to their atomicity [valence] the elements followed one another in the arithmetical sequence of their atomic weights." Dimitri Ivanovich Mcndeleyev (Mendeleev), 1869 "The periodicity in the properties of the elements is connected with the continuing build up and completion of the various electron groups that takes place with increasing atomic number." Niels Henrik David Bohr, 1923 (Nobel Prize in Physics in 1922 "for his services in the investigation of the structure of atoms and of the radiation emanating from them".) Multi-Electron Atoms electron spin Dirac spin quantum number, m s magnetic properties diamagnetic paramagnetic unpaired e - ferromagnetic aufbau (filling up) principle Pauli exclusion principle Hund's Rule 1A Atomic Electron Configurations 1 2 H He 1s 1 2A 3A 4A 5A 6A 7A Li 2s 1 Be exceptions to normal filling order B C N O F Ne Na Mg Al Si P S Cl Ar 3s 1 3B 4B 5B 6B 7B B B 2B K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 4s 1 3d 5 4s 1 3d 10 4s Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe 5s 1 4d 4 5s 1 4d 5 5s 1 4d 7 5s 1 4d 8 5s 1 4d 10 4d 10 5s Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn 6s 1 5d 9 6s 1 5d 10 6s Fr Ra Ac Rf Db Sg Bh Hs Mt Ds Rg Cn Fl Fv 7s 1 6d 10 7s Lanthanides Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 4f 1 d 1 s 2 4f 7 d 1 s 2 4f 14 d 1 s Actinides Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr 6 d 2 7s 2 5f 2 d 1 s 2 5f 3 d 1 s 2 5f 4 d 1 s 2 5f 7 d 1 s 2 5f 14 d 1 s 2 8A

4 - 4 - Aufbau (Building Up) Principle (15.8) EX. 1 For the element Pb a) Give the full electron configuration of its ground state. Hund b) Give the electron configuration of its ground state using the noble gas core. c) Give the orbital diagram of its valence electrons. Pauli d) Predict the more stable ions for Pb. octet in previous level Li Effective Nuclear Charge Shielding and Effective Nuclear Charge, ζ Li: 1s 2 2s charge that the 2s electron experiences when it is closer or further from the nucleus.

5 - 5 - shielding core electrons shield outer electrons from full effect of nuclear charge effective nuclear charge nuclear charge actually experienced by outer electrons due to shielding, decreases with increasing shielding penetration probability of electron being closer to nucleus in spite of shielding; order is s < p < d FIGURE 15.7 Periodic Trends (15.9) Size of Atoms and Ions In H10 homework the average value of r for a hydrogenic atom was given as n 2 a 0 3 Z l(l + 1) 2 2n 2 so its ~ n 2 a 0 /Z. In terms of the effective nuclear charge we have r ~ n 2 a 0 /ζ First Ionization Energies (IE) of the Elements: X(g) > X + (g) + e - IE energy of orbital from which e - came 1st IE removes highest energy e - (least bound); successive IEs higher in energy distinction between core and valence electrons seen in higher IE reveals FIGURE 15.7 shell, subshell structure linearity in IE across a period noble gas configuration noble gases very stable - octet good metals Group I least stable - give up e - to form octet effect of e - repulsions repulsive e - pairing energy

6 - 6 - Electron Affinities (EA) of the Elements X(g) + e - > X - (g) negative values when reaction is exothermic and anion is a more stable species than neutral atom no atom has negative EA for adding a second e - noble gases and most alkaline earths have positive EAs reveals shell, subshell structure noble gas configuration effect of e - repulsions FIGURE 15.7 Electronegativities (EN) of the Elements ability of an atom in a molecule to attract shared electrons to itself combines IE and EA into one property (one way to define) same trend as ionization energy - increases going up a group and left to right across a period EX 2. Arrange the following in order and explain your reasoning: a) Mg 2+, Ar, Br -, Ca 2+ in order of increasing radius b) Na, Na +, O, Ne in order of increasing ionization energy c) H, F, Al, O in order of increasing electronegativity EX 3. The first ionization energy for He is the highest for any element. Which element would have the highest second ionization energy? Explain. EX 4. Why is the electon affinity of Cl more negative than the electron affinity of S?

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