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5 QUANTUM MECHANICS AND ATOMIC STRUCTURE CHAPTER 5.1 The Hydrogen Atom 5.2 Shell Model for Many-Electron Atoms 5.3 Aufbau Principle and Electron Configurations 5.4 Shells and the Periodic Table: Photoelectron Spectroscopy 5.5 Periodic Properties and Electronic Structure General Chemistry I 1 Colors of Fireworks from atomic emission red from Sr orange from Ca yellow from Na green from Ba blue from Cu General Chemistry I 2 5.1 THE HYDROGEN ATOM - The hydrogen atom is the simplest example of a one-electron atom or ion (i.e. H, He+, Li2+, …). - For solution of the Schrödinger equation, Cartesian coordinates, x, y, z spherical coordinates, r, q, f to express angular orientation. General Chemistry I 3 Energy Levels For a hydrogen atom, V(r) = Coulomb potential energy Solutions of the Schrödinger equation E = En = _ Z 2e4me 8 0 n h 2 2 2 n = 1, 2, 3,..... 1 rydberg = 2.18×10-18 J Principal quantum number n indexes the individual energy levels. General Chemistry I 4 Coordinate Systems V(x, y, z) – can be expressed in various coordinate systems. Spherical polar coordinates (r, θ, Ф) vs. Cartesian coordinates (x, y, z) h 2 2 2 2 V ( x , y , z ) 2 2 ( x, y, z ) E ( x, y, z ) 2 2 8 m x y z For Coulomb potential, V, convenient to express in SPC (V = -e2/r); Cartesian, V = -e2/[x2 + y2 +z2]1/2 - Cartesian Coordinates h 2 2 2 2 e2 2 2 2 2 ( x, y, z ) E ( x, y, z ) 2 2 2 8 m x y z ( x y z ) - Spherical Polar Coordinates (SPC) h 2 1 2 1 2 2 r 2 sin q q 8 m r r r r sin q q General Chemistry I 1 2 e 2 ( x, y, z ) E ( x, y, z ) 2 2 2 r sin q f r 5 Quantization of the Angular Momentum any integral value from 0 to n-1 value of l 0 1 2 3 orbital type s p d f General Chemistry I orbitals 6 For every value of n, n2 sets of quantum numbers General Chemistry I 7 Energy level diagram for the H atom These sets of state are said to be degenerate. General Chemistry I 8 Boundary Conditions Boundary Conditions yield quantum numbers! General Chemistry I 9 Wave Functions radial part General Chemistry I angular part: spherical harmonics 10 - Spherical Polar Coordinates and the Wave Function h 2 1 2 1 2 2 r 2 sin q q 8 m r r r r sin q q 1 2 e 2 ( x, y, z ) E ( x, y, z ) 2 2 2 r sin q f r h 2 1 2 e 2 1 E ( x, y, z ) 2 2 2 r sin q q r sin q q 8 m r r r r 1 2 ( x, y , z ) 0 2 2 2 r sin q f h 2 2 e 2 r 2 1 2 r Er ( x , y , z ) sin q 2 sin q q r q 8 m r r 1 2 ( x, y , z ) 0 2 2 sin q f Depends only on r Depends only on q, f - Schrödinger wave function → factored into radial (R) and angular (Y) functions (r,q ,f ) R(r )Y (q ,f ) General Chemistry I 11 Orbitals EXAMPLE 5.1 General Chemistry I 12 Specification of the Wave Functions: Orbitals Schrödinger’s Wave Function i.e.) n = 3 (only one orbit in the Bohr- de Broglie model) 32 (nine) different ways the electron can vibrate to form a standing wave → 9 degenerate quantum states Orbitals replacement of Bohr’s orbit for 3D Schrödinger’s wave function - Radial part, Rnℓ(r) Laguerre polynomial (rn-1) x e-r/(na0 ) - Angular part, Yℓm(θ,Ф) Legendre function (sin θ & cos θ) x eimФ * a0 ← Bohr’s radius General Chemistry I 13 No radial node n=1 Spherical Symmetry: no angular nodes n = 2 1 radial node n = 3 2 radial nodes n = 2 No radial node 1 angular node n = 3 1 radial node 2 angular nodes n = 3 No radial node General Chemistry I 14 Sizes and Shapes of Orbitals Graphical representations of the orbitals 1) Slicing up 3D space into various 2D and 1D regions and examining the value of wave function at each point. General Chemistry I 15 2) Looking only at the radial behavior (“vertical slice”) R100(r) General Chemistry I 16 General Chemistry I 17 s orbitals s orbital: constant Y All s orbitals are spherically symmetric. General Chemistry I 18 - 1s (n = 1, ℓ = 0, m = 0) R10(r) and Y00(θ,Ф) a function of r only * spherically symmetrical * exponentially decaying * no nodes - 2s (n = 2, ℓ = 0, m = 0) R20(r) and Y00(θ,Ф) zero at r = 2a0 = 1.06Å nodal sphere or radial node [r<2ao: Ψ>0, positive] [r>2ao: Ψ<0, negative] General Chemistry I 19 2s 3s isosurface 1s A surface of points with the same value of wave functions wave function plot radial probability density (or function) plot General Chemistry I 20 Radial Distribution Functions: RDFs Prob(r) = r2[R(r)]2 2 f q 0 r 2 sin q dq df 2 0 4 r 2 4 r 2 R (r )Y (q , f ) 2 4 r 2 R (r ) Y (q ,f ) 2 4 r R (r ) 1/ 2 2 2 2 2 r 2 R (r ) 2 2 ┃Ψ┃2 x 4πr2 dr General Chemistry I 21 Boundary Surfaces - No clear boundary of an atom - Defining the size of atom as the extent of a “balloon skin” inside which 90% of the probability density of the electron is contained. 1s: 1.41 Å, 2s: 4.83 Å, 3s: 10.29 Å An ns orbital has n-1 radial nodes. General Chemistry I 22 p Orbitals 2p0 orbital : R21 Y10 (n = 2, ℓ =1, m = 0) 2 · Ф = 0 → cylindrical symmetry about the z-axis · R21(r) → r/a0 no radial nodes except at the origin · cos θ → angular node at θ = 90o, x-y nodal plane · r cos θ → z-axis 2p0 → labeled as 2pz General Chemistry I 23 2p+1 and 2p-1 orbitals: R21Y11, R21Y1-1 (n = 2, ℓ = 1, m = ±1) · Y11(θ,Ф) → e±iФ = cosФ ± i sinФ ← Euler’s formula was used · taking linear combinations of these gave two real orbitals, called 2px and 2py - px and py differ from pz only in the angular factors (orientations). General Chemistry I 24 Wave function and RDF plots for 2p and 3p orbitals 2p 3p r2Rnl2 General Chemistry I 25 General Chemistry I 26 d Orbitals d-orbitals General Chemistry I 27 General Chemistry I 28 Orbital Shapes and Sizes: General Notes General Chemistry I 29 General Chemistry I 30 Stern-Gerlach Experiment: experimental evidence for the existence of electron spin Stern and Gerlach (1926): Ground state Na atoms were passed through a strong inhomogeneous magnetic field. · All spin magnetisms in inner orbitals → cancelled, except an electron in 3s orbital. · Y (θ,Ф)(3s) → no rotation · The intrinsic magnetism of the 3s electron → only possible to respond to the external field. · 3s electrons → line up with their north poles either along or against the main north pole of the magnet. The lack of a “straight-through” beam with the magnet → clear evidence of twovalued electron’s magnetism. General Chemistry I 31 Electron Spin ms, spin magnetic quantum number - An electron has two spin states, as ↑(up) and ↓(down), or a and b. - the values of ms, only +1/2 and -1/2 General Chemistry I 32 5.2 SHELL MODEL FOR MANY-ELECTRON ATOMS - In many-electron atoms, Coulomb potential energy equals the sum of nucleus-electron attractions and electron-electron repulsions. - No exact solutions of Schrödinger equation - In a helium atom, r1 = the distance of electron 1 from the nucleus General Chemistry I r2 = the distance of electron 2 from the nucleus r12 = the distance between the two electrons 33 r1≡ (r1,θ1,Ф1) interelectron distance r12 ≡┃r1 – r2┃ r2≡ (r2,θ2,Ф2) Interparticle distance in He Repulsion between electrons: This new cross-term makes the Schrodinger equation impossible to solve exactly and causes the Bohr model to fail for He and other atoms Screening of one electron by another General Chemistry I (r1,q1,f1, r2 ,q2 ,f2 ) (r1, r2 ) 34 Hartree Orbitals Self-consistent field (SCF) orbital approximation method by Hartree, Fock, Slater, and others - all electrons were treated as independent, meaning that we neglect the repulsion term. - Building up an effective nuclear charge Zeffe (r1,q1,f1, r2 ,q2 ,f2 ) (r1, r2 ) a (r1 ) b (r2 ) i.e.) for He, Zeff(He) = 1.6875 < 2 General Chemistry I 35 Three simplifying assumptions by Hartree The wave function becomes a product of these one-electron orbitals. 3. The effective field is spherically symmetric; that is, it has no angular dependence. The equations for the unknown effective field and the unknown one-electron orbitals must be solved by iteration until a selfconsistent solution is obtained. General Chemistry I 36 E.g. for the ground state of He, 1 a0 3/ 2 Zr / a 1 a0 3/ 2 Zr / a (r1 , r2 ) a ( r1 ) b ( r2 ) e 1 0 e 2 0 Z Z 1 a 3/ 2 1 a Z r / a eff 1 0 0 e 0 Z eff Z eff Z eff 3 Zeff r1 / a0 Z eff r2 / a0 e e a 03 3/ 2 1s2 1s orbital General Chemistry I e Z eff r2 / a0 occupancy 1s orbital 37 The ground state of Ar, Z = 18 General Chemistry I 38 sum of the radial probability density functions of all the occupied orbitals. General Chemistry I 39 Shielding Effects - Energy-level diagrams for many-electron atoms 1) The degeneracy of the p, d, and f orbitals is removed, due to the difference of Zeff from the Coulomb field. 2) Energy values are distinctly shifted from the values of corresponding H orbitals due to the strong attraction by nuclei with Z > 1. - Each electron attracted by the nucleus, and repelled by the other electrons. → shielded from the full nuclear attraction by the other electrons. General Chemistry I 40 - effective nuclear charge, Zeff e < Ze - Hartree orbital energy (rydbergs) E.g. For Ar (Z =18), Zeff(1) ~ 16, Zeff(2) ~ 8, Zeff(3) ~ 2.5 General Chemistry I 41 Energy level diagram for many-electron atoms General Chemistry I 42 Penetration: s-electron – very close to the nucleus highly penetrates through the inner shell. p-electron – penetrates less than an s-orbital effectively shielded from the nucleus In a many-electron atom, because of the effects of penetration and shielding, the order of energies of orbitals in a given shell is s < p < d < f. Order of orbital shielding (for fixed value of n): Zeff(s) > Zeff(p) > Zeff(d) > Zeff(f) General Chemistry I 43 5.3 AUFBAU PRINCIPLE AND ELECTRON CONFIGURATIONS The Building-up (Aufbau) Principle is a set of rules that allows us to construct ground state electron configurations of the elements. 1. 2. 3. Assume electrons ‘occupy’ orbitals in such a way as to minimize the total energy (lowest energy first). Assume a maximum of two electrons can ‘occupy’ an orbital and these must have opposite spins. (Pauli’s Exclusion Principle: no two electrons in an atom can have the same set of quantum numbers). Assume electrons occupy unoccupied degenerate subshell orbitals first, and with parallel spins (Hund’s rule). In building-up electron configurations, in order of energy, subshell energy overlap must be taken into account (see later). General Chemistry I 44 Electron configurations of first ten elements (H to Ne) Electron configuration of an atom is a list of all its occupied orbitals, with the numbers of electrons that each one contains. Standard notation Box and arrow notation (showing electron spins) H, 1s1 He, 1s2 Li, 1s22s1 or [He]2s1 Valence electron Core electrons General Chemistry I 45 Be, 1s22s2 or [He]2s2 B, 1s22s22p1 or [He]2s22p1 C, 1s22s22p2 or [He]2s22p2 → 1s22s22px12py1 General Chemistry I parallel spins by Hund’s rule 46 - Elements in Period 2 (from Li to Ne), the valence shell with n = 2. - p-block elements: N to Ne, filling of p orbitals s-block elements: H to Be, filling of s orbitals General Chemistry I 47 Magnetic properties as a test of electronic configurations - paramagnetic: a substance attracted into a magnetic field with one or more unpaired electrons - diamagnetic: a substance pushed out of a magnetic field with all electrons paired Examples: paramagnetic: H, Li, B, C diamagnetic: He, Be C General Chemistry I 48 General Chemistry I 49 n = 3: Na, [He]2s22p63s1 or [Ne]3s1 to Ar, [Ne]3s23p6 n = 4: from Sc (scandium, Z = 21) to Zn (zinc, Z = 30) the next 10 electrons enter the 3d-orbitals (d-block elements) The (n+l) rule Order of filling subshells in neutral atoms is determined by filling those with the lowest values of (n+l) first. Subshells in a group with the same value of (n+l) are filled in order of increasing n, due to orbital screening. order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < … n = 5: 5s-electrons followed by the 4delectrons n = 6: Ce (cerium, [Xe]4f15d16s2) General Chemistry I 50 Alternative Pictorial Representation of Orbital Energy Order General Chemistry I 51 Anomalous Configurations Exceptions to the Aufbau principle General Chemistry I 52 Filling of electron subshells in relation to the periodic table General Chemistry I 53 All atoms in a given period have the same type of core with the same n. All atoms in a given group have analogous valence electron configurations that differ only in the value of n. Group 18/VIII Group 1/IA Period 2 Period 3 Period 4 Period 5 Period 6 Period 7 General Chemistry I 54 5.4 SHELLS AND THE PERIODIC TABLE: PHOTOELECTRON SPECTROSCOPY A shell is defined precisely as a set of orbitals that have the same principal quantum number. The photoelectron spectroscopy (PES) principle determining the energy level of each orbital by measuring the ionization energy required to remove each electron from the atom General Chemistry I 55 Basic design of a PE spectrometer General Chemistry I 56 General Chemistry I 57 - Koopmans’ approximation, - with the frozen orbital approximation: the orbital energies are the same in the ion, despite the loss of an electron. E.g. for Ne with 1s2 2s2 2p6 General Chemistry I 58 5.28 Photoelectron spectroscopic studies of silicon atoms excited by X-rays of wavelength 9.890 x 10-10 m show four peaks in which the electrons have speeds (all x 107 m/s) of (1) 2.097, (2) 2.093, (3) 2.014, and (4) 1.971. (a) Calculate the ionization energy of the electrons corresponding to each peak. (b) Assign each peak to an orbital of the silicon atom. 1eV = 1.6022 x 10-19 J; c = 2.99792 x 108 m/s; h = 6.62607 x 10-34 J s; me = 9.10938 x 10-31 kg Solutions (a) E(X-rays) = h = hc/ = (6.62607 x 10-34 J s)(2.99792 x 108 m/s) 9.890 x 10-10 m = 2.0085 x 10-16 J = (2.0085 x 10-16 J) 1 eV 1.6022 x 10-19 J = 1253.6 eV Using the PES equation, IE = h - KE = h - 1/2mev2, 1 eV (9.10938 x 10-31 kg)(2.097 x 107 m/s)2 _ IE(1) = 1253.6 eV 1.6022 x 10-19 J 2 = 3.5 eV General Chemistry I 59 IE(2) =1253.6 eV _ (9.10938 x 10-31 kg)(2.093 x 107 m/s)2 2 1 eV 1.6022 x 10-19 J = 8.3 eV IE(3) =1253.6 eV _ (9.10938 x 10-31 kg)(2.014 x 107 m/s)2 2 1 eV 1.6022 x 10-19 J = 100.5 eV IE(4) =1253.6 eV _ (9.10938 x 10-31 kg)(1.971 x 107 m/s)2 2 1 eV 1.6022 x 10-19 J = 149.2 eV (b) The ground-state electron configuration of silicon is 1s22s22p63s23p2. Peak 1 corresponds to removal of silicon’s 3p electrons, which are its least tightly bound electrons. Peaks 2, 3, and 4 correspond to removal of Si’s 3s, 2p, and 2s electrons respectively. The 1s electron does not give a peak. It is probably at an energy lower than –1253.6 eV and inaccessible with the X-radiation used in this experiment. General Chemistry I 60 (Fig. 5.25) Energies of atomic subshells, determined by PES General Chemistry I 61 5.5 PERIODIC PROPERTIES AND ELECTRONIC STRUCTURE Atomic radius: defined as half the distance between the centers of neighboring atoms Ionic radius: its share of the distance between neighboring ions in an ionic solid General trends - r decreases from left to right across a period (effective nuclear charge increases) - r increases from top to bottom down a group (change in n and size of valence shell) General Chemistry I 62 - The rate of increase changes considerably. i.e. Li+, Na+, to K+ substantial change to Rb+, Cs+ small change due to filling d-orbitals Lanthanide contraction filling of the 4f orbitals (poor shielders) radii of 3rd row D-block elements similar to those of 2nd row General Chemistry I 63 Molar volumes (cm3 mol-1) of atoms in the solid phase = size of the atoms + geometry of the bonding General Chemistry I 64 Periodic Trends in Ionization Energies From left to right, generally increase in IE1 due to the increase of Zeff From top to bottom, generally decrease in IE1 due to the increase of n From He to Li, a large reduction in IE1 2s e- further than 1s e-, and 2s e- sees a net +1 charge From Be to B, slight reduction in IE1 fifth e- in a higher energy 2p orbital From N to O, slight reduction in IE1 2 e- in the same 2p orbital leading to greater repulsion General Chemistry I 65 General Chemistry I 66 Electron Affinity - The periodic trends in EA parallel those in IE1 with one unit lower shift e.g. F to F-, large EA because of its closed-shell configuration (kJ mol-1) General Chemistry I 67 10 Problem Sets For Chapter 5, 6, 18, 22, 28, 32, 38, 44, 46, 48, 56 General Chemistry I 68