Atomic Orbitals 1 2 Atomic Orbitals • What is an orbital? Miessler and Tarr, Chapter 2 – An electron wave, a matter wave • How is an orbital described? – Three quantum numbers Number Value Property n 1, 2, 3, … Orbital size, E of e- in H atom l (“ell”) 0, 1, 2 …(n-1) Orbital shape ml - l ..0..+ l Orbital orientation 3 Where do QN Come From? • Schrodinger eqn. For e- wave in 3 dimensions !2 " !x2 + ! 2" !y 2 + ! 2" !z2 + 8π 2m h2 4 ! usually obtained in polar coordinates z (E − V)" = 0 r ! Where E = total energy and V = potential energy m = electron mass and h = Planck’s constant y Solution to equation gives only certain values: 1. For E = eigenvalues 2. For ! = eigenfunctions or wave functions This introduces the quantum condition naturally Wave Functions ! (r, ", #) = [Rn,l (r) ] ["l,m #m] Radial Spherical harmonic " x 5 Wave Functions 6 ! (r, ", #) = [Rn,l (r) ] ["l,m #m] Radial Spherical harmonic = Y l,m (",#) Wave functions are normalized. The wave function must lead to a probability of finding an electron somewhere in space of 1. " = Zr/ao where ao = 52.9 pm 1 Wave Functions 7 ! (r, ", #) = [Rn,l (r) ] ["l,m #m] 8 ! (r, ", #) = [Rn,l (r) ] ["l,m #m] Spherical harmonic Radial Wave Functions Radial Spherical harmonic Consider radial portion— • n cannot be 0 because R would then be 0 • n - l - 1 restricts values of l to n-1 – n = 1, then l = 0 – n = 2, then l = 0, 1 – n = 3, then l = 0, 1, 2 Wave Functions 9 ! (r, ", #) = [Rn,l (r) ] ["l,m #m] Radial = Y l,m (",#) = Y l,m (",#) l - |m| sets up the condition that values of m l are - l … 0 …+ l ! 1$ R1,0 = 2 # & " a0 % • • • • 3/2 Spherical harmonic $m portion is imaginary Therefore, ! is itself meaningless, but !2 does have a physical interpretation #l,m portion: 2 l + 1 = number of values of m l For n = 1 and l = 0 1s atomic orbital 10 ! (r, ", #) = [Rn,l (r) ] ["l,m #m] Spherical harmonic Radial Wave Functions !2 proportional to WAVE INTENSITY or PROBABILITY When l = 0 there is no dependence on angle. 11 Radial Wave Functions 12 e-r/a R has units of (1/pm)3/2 R2 proportional to (1/pm)3 Therefore, R2 = density function R2 proportional to probability of finding electron in a given volume with radius r. 2 13 For n = 1 and l = 0 1s atomic orbital ! 1$ R1,0 = 2 # & " a0 % • • • • • 3/2 ! 1$ R1,0 = 2 # & " a0 % e-r/a R21,0 = (4/ao3)e-2r/a When r = 0, R21,0 = 4/ao3 = 2.7 x 10-5 /pm3 Because Y0,0 = (√2/2)(1/√2 π) = 1/2√π R2Y2 for 1s = 6.8 x 10-7 /pm3 This is the probability density for a 1s electron essentially at the nucleus. • A NODE is a plane or point of 0 electron probability. • s orbitals all have finite probability density “at” the nucleus. • p, d, and f orbitals have 0 density “at” the nucleus. That is, there is a node passing thru the nucleus. • All orbitals can have nodes in the electron wave beyond the nucleus. orbital 1 0.6 Series1 0.4 0.2 0 2 3 Distance from 4 5 16 Nodes in Electron Waves 0.8 1 e-r/a • It is important to recognize that s orbitals (l = 0) have probability density greater than 0 “at” the nucleus. 1.2 0 3/2 15 1s Atomic Orbital ! (= e-r) vs. r 1s 14 For n = 1 and l = 0 1s atomic orbital 6 nucleus 17 18 2s Atomic Orbital ! [= (2-r)e-r/2] vs. r 2s Atomic Orbital !2 vs. r 2s Orbital 2s Wave Function Squared 2.5 0.4 0.35 2 0.3 1.5 0.25 Series1 1 0.2 Series1 0.15 0.5 0.1 0.05 0 0 2 4 6 8 10 12 0 0 -0.5 Distance from nucleus 2 4 Distance 6 from 8 10 12 nucleus 3 19 2p Atomic Orbital ! ( = r·e-r/2) vs. r 20 Radial Wave Functions A Summary R = (const) (eqn in !) !l e-!/n 2p Atomic Orbital 0.8 0.7 0.5 0.4 Series1 Larger n makes R larger. If l not 0 there is a node at nucleus. No. of radial nodes = n-l-1 0.6 0.3 0.2 0.1 l = number of nodal planes 0 0 2 4 6 Distance from 8 the 10 12 nucleus Number of radial nodes = n - l - 1 Angular Functions/Orbital Shapes Consider pz orbital: Y = 3 4π " z$ = # r% z Note that the number of planar nodes = l Wave function is + in +z direction 21 3 cos & 4π Consider dxy orbital: cos ! = cos (0˚) = 1 a) When either x or y is 0, then Y = 0 and there is a node. b) When either x or y is negative, then Y has negative value. c) When both x and y are negative, Y has a positive value. cos (90˚) = 0 x Planar node Wave function is - in -z direction Angular Functions/Orbital Shapes Y= 60 " x y% $ ' 16π # r 2 & nodal plane -x, -y -x, +y + – – + nodal plane cos (180˚) = -1 y +x, -y x +x, +y 23 Polyelectronic Atoms 22 24 Effective Nuclear Charge, Z* • In polyelectronic atoms, orbital energies increase in the order (n+ l) • For two orbitals of same (n+l), the one with smaller n lies lower in energy. (n+l) 1 2 3 1s 2s 2p,3s 4 3p, 4s 5 3d, 4p, 5s 6 4d, 5p, 6s Why should 2s lie lower in energy than 2p in a polyelectronic atom? Shielding ---> effective nuclear charge, Z*. Electron cloud for 1s electrons 4 Slater’s Rules Calculating Z* Z* = Z - % where % is the shielding constant Electrons of higher n have no effect Electrons of same n, % = 0.35/electron Electrons of n - 1, % = 0.85/electron Electrons of lower than n - 1, % = 1.00 For Li 2s electron: % = 2 x 0.85 = 1.70 Z* = 3.00 - 1.70 = 1.30 25 Calculated Effective Nuclear Charges, Z* B 2.60 Al 3.50 Ga 5.00 In 5.00 Tl 5.00 C 3.25 Si 4.15 Ge 5.65 Sn 5.65 Pb 5.65 N 3.90 P 4.80 As 6.30 Sb 6.30 Bi 6.30 O 4.55 S 5.45 Se 6.95 Te 6.95 Po 6.95 F 5.20 Cl 6.10 Br 7.60 I 7.60 At 7.60 26 Ne 5.85 Ar 6.75 Kr 8.25 Xe 8.25 Rn 8.25 5