1. Lecture 2 THE ELECTRONIC STRUCTURE OF THE POLYELECTRONIC ATOM. PART I 1) Theoretical description of the polyelectronic atom: Schrödinger equation looks the same: H x-electron  = E  But: Compare Hamiltonian operator for hydrogen atom with a single electron: H 1-electron And that for polyelectronic atom with x electrons (spin-orbit coupling is neglected): H x-electron Schrödinger equation for polyelectronic atom cannot be solved exactly since electron-electron distances r ij and thus the last term of the operator H x-electron are unknown before the equation is solved For the case of the polyelectronic atom the Schrödinger equation can be solved approximately if we assume that each of x electrons is independent of others (one-electron approximation):  =  (1)  (2) …  (x), E = E(1)+E(2)+…+E(x)

2. 2) Hydrogen-like orbitals Resulting one-electron wave functions give rise to so-called hydrogen-like orbitals They are of the same shape as corresponding hydrogen atom orbitals, but Contracted. With increased nuclear charge Z size of atoms decreases along the row of elements with increased Z. So, for 1s orbitals R(r) = Their energy is a function of both n and l. One-electron energy increases in the order ns < np < nd < nf. The reason for that is electron-electron interactions. (Compare with E = for the hydrogen atom orbitals) The orbital energy ordering is therefore complex. Energy of the orbitals with quantum number n may be higher than that of the orbitals with quantum number n+1 or even n+2, i.e. 3d > 4s, 4f > 6s etc. There is no uniform ordering of orbital energies for all elements. The following series is very useful though there are exceptions: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 5d ≈ 4f < 6p < 7s < 6d < 5f

4. 3) Electron spin and Pauli principle Each electron has an angular momentum. (An experimental evidence for electron spin: next lecture). Spin quantum number m s is equal to the value of the projection of the electron-spin angular momentum on z-axis and is expressed in units of ħ. m s can have one of two values only, +½ or -½. The fourth, spin quantum number, m s (+½ and -½) is therefore needed to fully characterize the atomic structure. The electrons with identical set of four quantum numbers, n,l, m l, m s cannot be found in a given atom (Pauli exclusion principle) Since there are two allowed values of spin quantum number m s, +½ and -½, each orbital can be occupied by two electrons with opposite spins (one  - and one  - electron). Therefore for a given n the total number of electrons is 2n 2. Each electron has also magnetic moment. The magnitude of the magnetic moment  is given by  = 2.00[s(s+1)] 1/2 (here s is spin, s = |m s |) expressed in Bohr magnetons (1 Bohr magneton = 9.27·10 ‑ 24 A m 2 ).

5. 4) The Aufbau principle To find the electron distribution (electron configuration) of a polyelectron atom, the Aufbau principle formalism is used. According to it, when protons (and neutrons) are added to the nucleus, the electrons are added to orbitals in the sequence in which energies of hydrogen-like orbitals increase: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 5d ≈ 4f < 6p < 7s < 6d < 5f Electronic configuration of the elements: http://chemistry.about.com/library/weekly/aa013103a.htmhttp://chemistry.about.com/library/weekly/aa013103a.htm ElementConfigurationElement Configuration H1s 1 He1s 2 Li[He]2s 1 Na[Ne]3s 1 K[Ar]4s 1 Be[He]2s 2 Mg[Ne]3s 2 Ca[Ar]4s 2 B[He]2s 2 2p 1 Al[Ne]3s 2 3p 1 Sc[Ar]4s 2 3d 1 C[He]2s 2 2p 2 Si[Ne]3s 2 3p 2 Ti[Ar]4s 2 3d 2 N[He]2s 2 2p 3 P[Ne]3s 2 3p 3 V[Ar]4s 2 3d 3 O[He]2s 2 2p 4 S[Ne]3s 2 3p 4 F[He]2s 2 2p 5 Cl[Ne]3s 2 3p 5 Ne[He]2s 2 2p 6 Ar[Ne]3s 2 3p 6

6. 5) Filled shells and relativistic effects There are no exceptions from the rule for the elements H-V. Some exceptions which occur for Cr and some other heavier elements (Cu, Nb, Mo, Ru-Ag and also 4f and 5f-block elements) are due to: a) the increased stability of half- or completely filled d-shells (Cr, Cu, Mo, Pd, Ag); b) the relativistic radial expansion and energetic destabilization of the outer 5d-, 6d and 5f-shells (i.e. compare Ta – Pt with Nb – Pd). ElementConfigurationElement Configuration V[Ar]4s 2 3d 3 As [Ar]4s 2 3d 10 4p 3 Tc[Kr]5s 2 4d 5 Cr[Ar]4s 1 3d 5 Se [Ar]4s 2 3d 10 4p 4 Ru[Kr]5s 1 4d 7 Mn[Ar]4s 2 3d 5 Br [Ar]4s 2 3d 10 4p 5 Rh[Kr]5s 1 4d 8 Fe[Ar]4s 2 3d 6 Kr [Ar]4s 2 3d 10 4p 6 Pd[Kr]4d 10 Co[Ar]4s 2 3d 7 Rb[Kr]5s 1 Ag[Kr]5s 1 4d 10 Ni[Ar]4s 2 3d 8 Sr[Kr]5s 2 Cd[Kr]5s 2 4d 10 Cu[Ar]4s 1 3d 10 Y[Kr]5s 2 4d 1 … Zn[Ar]4s 2 3d 10 Zr[Kr]5s 2 4d 2 Ta[Xe]4f 14 6s 2 5d 3 Ga [Ar]4s 2 3d 10 4p 1 Nb[Kr]5s 1 4d 4 W[Xe]4f 14 6s 2 5d 4 Si [Ar]4s 2 3d 10 4p 2 Mo[Kr]5s 1 4d 5 Re[Xe]4f 14 6s 2 5d 5

7. Summary The Schrödinger equation for multyelectron atoms cannot be solved directly. One-electron approximation and SCF procedure are used usually to solve it Resulting hydrogen-like orbitals are contracted and follow a different energy order as compared with hydrogen orbitals Fourth quantum number m s is required to completely describe multielectron systems. Pauli exclusion principle is applied The Aufbau principle allows to determine (approximate) electronic configuration of polyelectronic atoms; exceptions are due to increased stability of half- or completely filled shells and the relativistic radial expansion and energetic destabilization of the outer 5d- and 5f-shells