Chapter 21 Multi-Electron Atoms

P. J. Grandinetti

Chem. 4300

P. J. Grandinetti Chapter 21: Multi-electron Atoms The Helium Atom To a good approximation, we separate wave function into 2 parts: one for center of mass and other for 2 electrons orbiting center of mass. Hamiltonian for 2 electrons of He-atom is

ℏ2 ℏ2 Zq2 Zq2 q2 ̂ = − ∇2 − ∇2 − e − e + e He 2m 1 2m 2 4휋휖 r 4휋휖 r 4휋휖 r y ⏟⏞⏟⏞⏟e ⏟⏞⏟⏞⏟e ⏟⏞⏟⏞⏟0 1 ⏟⏞⏟⏞⏟0 2 ⏟⏞⏟⏞⏟0 12 ̂ ̂ ̂ ̂ ̂ KH (1) KH (2) VH (1) VH (2) Ve-e x

̂ ̂ 1st and 2nd terms on left, KH(1) and KH(2), are kinetic energies for 2 electrons ̂ ̂ − 3rd and 4th terms, VH(1) and VH(2), are Coulomb attractive potential energy of each e to nucleus ̂ 5th term inside brackets, Ve-e, is electron-electron Coulomb repulsive potential energy. ̂ It is Ve-e that makes ﬁnding analytical solution more complicated. ̂ Ve-e is not small perturbation compared to other terms in Hamiltonian, so we cannot ignore it.

P. J. Grandinetti Chapter 21: Multi-electron Atoms ̂ − − What happens if we ignore Ve-e, the e –e repulsion term? We have non-interacting electrons. Consider solutions we get when repulsive term is removed

̂ ̂ ̂ ̂ ̂ ̂ ̂ He = KH(1) + VH(1) + KH(2) + VH(2) = H(1) + H(2) ̂ − This He describes two non-interacting e s bound to He nucleus, so electron wave function is a product of single e− wave functions:

휓 휓 휓 He = H(1) H(2) Orbital approximation

Total energy using single e− wave functions is ( ) Z2 Z2 E = −(1 Ry) + 2 2 n1 n2

For both electrons in n1 = n2 = 1 states of He atom obtain E = -108.86 eV Neglecting repulsion gives ∼ 38% error compared to measured He ground state energy of −79.049 eV.

P. J. Grandinetti Chapter 21: Multi-electron Atoms The Variational Method

P. J. Grandinetti Chapter 21: Multi-electron Atoms The Variational Method 휓 Make a mathematical guess for ground state wave function, guess. Is it a good guess? 휓 If we choose a well behaved and normalized wave function, guess, then it can be expressed as a linear 휓 combination of exact eigenstates, n, ∑∞ 휓 휓 guess = cn n n=0 휓 휓 Exact eigenstates, n, and coeﬃcients, cn, needed to describe guess are unknown. 휓 Since guess is normalized we have ∑∞ ∑∞ 휓∗ 휓 휏 ∗ 휓∗ 휓 휏 ∫ guess guessd = 1 = cmcn ∫ m nd V n=0 m=0 V ∫ 휓∗ 휓 휏 ≠ Exact eigenstates are orthonormal, so V m nd = 0 when m n. Thus we know ∑∞ ∑∞ | |2 휓∗휓 휏 | |2 cn ∫ n nd = cn = 1 n=0 V n=0 | | ≤ Each cn is restricted to cn 1. 휓 휓 휓 If guess was perfect then guess = 0 with c0 = 1 and cn≠0 = 0 P. J. Grandinetti Chapter 21: Multi-electron Atoms The Variational Method Back to guess and calculate expansion ∑∞ ∑∞ 휓∗ ̂ 휓 휏 ∗ 휓∗ ̂ 휓 휏 ∫ guess guessd = cmcn ∫ m nd V n=0 m=0 V 휓 ̂ ∫ 휓∗ ̂ 휓 휏 ≠ Again, n are exact eigenstates of , so V m nd = 0 when m n. ∑∞ ∑∞ 휓∗ ̂ 휓 휏 | |2 휓∗휓 휏 | |2 ∫ guess guessd = cn En ∫ n nd = cn En V n=0 V n=0 휓 휓 Hope is that guess ≈ 0, with energy E0. Rearranging ∑∞ ( ) ∑∞ ∑∞ ∑∞ 휓∗ ̂ 휓 휏 | |2 | |2 | |2 | |2 ∫ guess guessd = cn E0 + (En − E0) = cn E0+ cn (En−E0) = E0+ cn (En−E0) V n=0 n=0 n=0 n=0 | |2 Since both cn and (En − E0) can only be positive numbers we obtain 휓∗ ̂ 휓 휏 ≥ ∫ guess guessd E0 V

P. J. Grandinetti Chapter 21: Multi-electron Atoms The Variational Method This is called the variational theorem.

휓∗ ̂ 휓 휏 ≥ ∫ guess guessd E0 V

It tells us that any variation in guess that gives lower expectation value for ⟨̂ ⟩ brings us closer to correct ground state wave function and ground state energy. Suggests algorithm of ▶ , , , 휓 , , , , ⃗ , ⃗ , varying some parameters, a b c …, in guess(a b c … x1 x2 …) and ▶ searching for minimum in

휓∗ , , , , ⃗ , ⃗ , ̂ 휓 , , , , ⃗ , ⃗ , 휏 ∫ guess(a b c … x1 x2 …) guess(a b c … x1 x2 …)d V

▶ 휓 , , , , ⃗ , ⃗ , to get best guess for guess(abest bbest cbest … x1 x2 …).

P. J. Grandinetti Chapter 21: Multi-electron Atoms Example Apply variational method to trial wave function for He atom making orbital approximation,

휓 휓 휓 He = H(1) H(2) ( ) ( ) ′ 3∕2 ′ 3∕2 1 Z ′ 1 Z ′ 휓 −Z r1∕a0 휓 −Z r2∕a0 using H(1) = √ e and H(2) = √ e 휋 a0 휋 a0 and use Z′ as variational parameter. Idea for replacing atomic number, Z, by Z′ is that each electron sees smaller eﬀective nuclear charge due to “screening” by other electron.

Solution: Recalling ℏ2 Zq2 ℏ2 Zq2 q2 ̂ = − ∇2− e − ∇2− e + e He 1 휋휖 2 휋휖 휋휖 2me 4 0r1 2me 4 0r2 4 0r12 [ ] [ ] [ ] ℏ2 Zq2 ℏ2 Zq2 q2 ⟨̂ ⟩ = 휓∗ (1) − ∇2 − e 휓 (1)d휏 + 휓∗ (2) − ∇2 − e 휓 (2)d휏 + 휓∗ (1)휓∗ (2) e 휓 (1)휓 (2)d휏 ∫ H 1 휋휖 H ∫ H 2 휋휖 H ∫ H H 휋휖 H H 2me 4 0r1 2me 4 0r2 4 0r12 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ ̂ ̂ ̂ H (1) H (2) Ve-e Deﬁne ΔZ = Z − Z′ (or Z = Z′ + ΔZ) ... P. J. Grandinetti Chapter 21: Multi-electron Atoms Variational Method Example Expand and rearrange 1st and 2nd terms taking ΔZ = Z − Z′ (or Z = Z′ + ΔZ) [ ] ℏ2 Z′q2 ΔZq2 ⟨̂ ⟩ = 휓∗ (1) − ∇2 − e 휓 (1)d휏 − 휓∗ (1) e 휓 (1)d휏 ∫ H 1 휋휖 H ∫ H 휋휖 H 2me 4 0r1 4 0r1 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 2nd [ 1st ] [ ] ℏ2 Z′q2 ΔZq2 q2 + 휓∗ (2) − ∇2 − e 휓 (2)d휏 − 휓∗ (2) e 휓 (2)d휏 + 휓∗ (1)휓∗ (2) e 휓 (1)휓 (2)d휏 ∫ H 2 휋휖 H ∫ H 휋휖 H ∫ H H 휋휖 H H 2me 4 0r2 4 0r2 4 0r12 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 3rd 4th 5th 1st and 3rd terms are H-atom energies except with Z′. We know these solutions. For 2nd and 4th terms we have ( ) 2 2 ′ 3 ∞ 2 ′ ΔZqe ΔZqe 1 Z ′ ΔZqe Z − 휓∗ (1) 휓 (1)d휏 = − r−1e−2Z r∕a0 4휋r2dr = − ∫ H 휋휖 H 휋휖 휋 ∫ 휋휖 4 0r 4 0 a0 0 4 0 a0

5th term is more work and I’ll leave it[ as an exercise] to show that 2 ′ 2 q 5Z q 1 휓∗ (1)휓∗ (2) e 휓 (1)휓 (2)d휏 = e ∫ H H 휋휖 H H 휋휖 4 0r12 4 0 8a0

P. J. Grandinetti Chapter 21: Multi-electron Atoms Variational Method[ ] Example[ ] Z′2q2 Z′ΔZq2 5Z′q2 [ ( ) ] q2 ⟨̂ ⟩ e e e 1 ′2 5 ′ e = −2 휋휖 − 2 휋휖 + 휋휖 = Z − 2 Z − Z 휋휖 8 0a0 4 0a0 4 0 8a0 16 4 0a0 ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏟ ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ 1st, 3rd 2nd,4th 5th Now comes minimization step where we calculate d⟨̂ ⟩∕dZ′ = 0 as ̂ [ ( ) ] 2 d⟨⟩ 5 q = 2 Z′ − Z − Z′ e = 0 which gives ΔZ = Z − Z′ = 5∕16. ′ 휋휖 dZ 16 4 0a0

( ) 2 5 2 q Thus we find ⟨̂ ⟩ e ≥ = − Z − 휋휖 E0 16 4 0a0 For He (Z = 2) we find Z′ = Z − ΔZ = 2 − 5∕16 = 1.6875, the effective nuclear charge after screening by other e−. For ground state energy this leads to

( ) 2 2 27 2 q q ⟨̂ ⟩ e . e . eV, = − 휋휖 = −2 85 휋휖 = −77 5 16 4 0a0 4 0a0 error is only 1.9% compared to measured He ground state energy of −79.049 eV. P. J. Grandinetti Chapter 21: Multi-electron Atoms Making the Orbital Approximation Good

The orbital approximation — the idea that electrons occupy individual orbitals — is not something chemists would give up easily.

In orbital approximation we write He atom ground state as

휓 1 휓 휓 훼 훽 훼 훽 = √ 1s(1) 1s(2) [ (2) (1) − (1) (2)] 2

But now you know that electron–electron repulsion term spoils this picture, making it only an approximation.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Mean ﬁeld potential energy In 1927 Douglass Hartree proposed an approach for solving multi-electron Schrödinger Eq that makes orbital approximation reasonably good. In Hartree’s approach each single electron is presumed to move in combined potential ﬁeld of

nucleus, taken as point charge Zqe, and all other electrons, taken as continuous negative charge distribution.

| |2 2 N 2 |휓 ⃗ | Zq ∑ q | j(r)| ⃗ e e 3 ⃗′ Vi(r) ≈ − 휋 + 휋휖 ∫ | | d r 4 r 4 V |⃗ ⃗′| j=1,j≠i 0 |r − r |

⃗ Vi(r) is “mean field” potential energy for ith electron in multi-electron atom. Details of Hartree’s approach and later refinements are quite complicated. For now we use it to go forward with confidence with the orbital approximation.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Douglas Hartree with Mathematician Phyllis Nicolson at the Hartree Diﬀerential Analyser at Manchester University in 1935.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Making the Orbital Approximation Good Consequence of Hartree’s successful solution is that chemists think of atoms and molecules in terms of single electrons moving in eﬀective potential of nuclei and other electrons.

For given electron of interest effective charge, Zeff, seen by this electron is 휎 Zeff = Z − Z is full nuclear charge, and 휎 is shielding by average # of e−s between nucleus and ith e−.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Making the Orbital Approximation Good

휎 Zeﬀ = Z −

In multi-electron atom 휎 increases with increasing 퓁.

▶ This is because electrons in s orbital have greater probability of being near nucleus than p orbital, so electron of interest in s orbital sees greater fraction of nuclear charge than if it was a p orbital.

▶ Likewise when electron of interest is in p orbital it sees greater fraction of nuclear charge than if it was in d orbital.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Energy of ith electron varies with both n and 퓁

Distinguish electrons with same n and 퓁 value as being in same subshell. Ground state electron conﬁguration is obtain by ﬁlling orbitals one electron at a time, in order of increasing energy starting Energy with lowest energy. Electrons in outer most shell of atom are valence electrons, while electrons in inner closed shells are core electrons.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Periodic Trends and Table Before QM there were many attempts to arrange known elements so that there were some correlations between their known properties. in 1869 Dimitri Mendeleev made ﬁrst reasonably successful attempt by arranging elements in order of increasing atomic mass, and, most importantly, found that elements with similar chemical and physical properties occurred periodically. He placed these similar elements under each other in columns. In 1914 Henry Moseley determined it was better to order elements by increasing atomic number, giving us periodic table we have today. Periodic table is arrangement of elements in order of increasing atomic number placing those with similar chemical and physical properties in columns. Vertical columns are called groups. Elements within a group have similar chemical and physical properties. Groups are designated by numbers 1-8 and by symbols A and B at top. Horizontal rows are called periods, designated by numbers on left. Two rows placed just below main body of table are inner transition elements.

P. J. Grandinetti Chapter 21: Multi-electron Atoms IA VIIIA 1.008 4.003

1H 2He 1 Hydrogen IIA IIIA IVA VA VIA VIIA Helium 6.941 9.012 10.81 12.011 14.007 15.999 18.998 20.179

3Li 4Be 5B 6C 7N 8O 9F 10Ne 2 Lithium Berylium Boron Carbon Nitrogen Oxygen Fluorine Neon 22.990 24.305 26.98 28.09 30.974 32.06 35.453 39.948 VIIIB 3 11Na 12Mg IIIB IVB VB VIB VIIB IB IIB 13Al 14Si 15P 16S 17Cl 18Ar Sodium Magnesium } Aluminum Silicon Phosphorus Sulfur Chlorine Argon 39.098 40.08 44.96 47.88 50.94 52.00 54.94 55.85 58.93 58.69 63.546 65.38 69.72 72.59 74.92 78.96 79.904 83.80

19K 20Ca 21Sc 22Ti 23V 24Cr 25Mn 26Fe 27Co 28Ni 29Cu 30Zn 31Ga 32Ge 33As 34Se 35Br 36Kr 4 Potassium Calcium Scandium Titanium Vanadium Chromium Manganese Iron Cobalt Nickel Copper Zinc Gallium Germanium Arsenic Selenium Bromine Krypton 85.47 87.62 88.91 91.22 92.91 95.94 (98) 101.1 102.91 106.4 107.87 112.41 114.82 118.69 121.75 127.60 126.90 131.29

37Rb 38Sr 39Y 40Zr 41Nb 42Mo 43Tc 44Ru 45Rh 46Pd 47Ag 48Cd 49In 50Sn 51Sb 52Te 53I 54Xe 5 Rubidium Strontium Yttrium Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium Palladium Silver Cadmium Indium Tin Antimony Tellurium Iodine Xenon 132.91 137.33 138.91 178.49 180.95 183.85 186.2 190.2 192.2 195.08 196.97 200.59 204.38 207.2 208.98 (244) (210) (222)

55Cs 56Ba 57La 72Hf 73Ta 74W 75Re 76Os 77Ir 78Pt 79Au 80Hg 81Tl 82Pb 83Bi 84Po 85At 86Rn 6 Cesium Barium Lanthanum Hafnium Tantalum Tungsten Rhenium Osmium Iridium Platinium Gold Mercury Thallium Lead Bismuth Polonium Astatine Radon (223) 226.03 227.03

87Fr 88Rd 89Ac 7 Francium Radium Actinium

140.12 140.9077 144.24 (145) 150.36 151.96 157.25 158.93 162.50 164.93 167.26 168.93 173.04 174.97

Lanthanide Series 58Ce 59Pr 60Nd 61Pm 62Sm 63Eu 64Gd 65Tb 66Dy 67Ho 68Er 69Tm 70Yb 71Lu Cerium Praseodymium Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium Erbium Thullium Ytterbium Lutetium 232.04 231.0359 238.03 237.05 (244) (243) (247) (247) (251) (254) (257) (258) (259) (260)

Actinide Series 90Th 91Pa 92U 93Np 94Pu 95Am 96Cm 97Bk 98Cf 99Es 100Fm 101Md 102No 103Lr Thorium Protactinium Uranium Neptunium Plutonium Americium Curium Berkelium Californium Einsteinium Fermium Mendelevium Nobellium Lawrencium

P. J. Grandinetti Chapter 21: Multi-electron Atoms Atomic radii

Period Period Period Period Period 1 2 3 4 5 He Ne Ar Kr

250

200

150

100 atomic radii/pm

0 0 He Be N 10 Mg 20 30 40 50

1 Size generally decreases across period from left to right because outermost valence electrons are more strongly attracted to the nucleus. ▶ Moving across period inner electrons ability to cancel increasing nuclear charge diminishes ▶ Inner electrons in s orbitals shield nuclear charge better than those in p orbitals. ▶ Inner electrons in p orbitals shield nuclear charge better than those in d orbitals, and so forth.

2 Size increases down a group. Increasing principal quantum number, n, of valence orbitals means larger orbitals and an increase in atom size.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Ionization Energy Energy required to remove an e− from gaseous atom or ion is called ionization energy. − First ionization energy, I1, is energy required to remove highest energy e from neutral gaseous atom. → + − . Na(g) Na (g) + e I1 = 496 kJ/mol Ionization energy is positive because it requires energy to remove e−. − Second ionization energy, I2, is energy required to remove 2nd e from singly charged gaseous cation. + → 2+ − . Na (g) Na (g) + e I2 = 4560 kJ/mol Second ionization energy is almost 10 times that of 1st because number of e−s causing repulsions is reduced.

Third ionization energy, I3, is energy required to remove 3rd electron from doubly charged gaseous cation. 2+ → 3+ − . Na (g) Na (g) + e I3 = 6913 kJ/mol

P. J. Grandinetti Chapter 21: Multi-electron Atoms Ionization Energy

→ + − . Na(g) Na (g) + e I1 = 496 kJ/mol + → 2+ − . Na (g) Na (g) + e I2 = 4560 kJ/mol 2+ → 3+ − . Na (g) Na (g) + e I3 = 6913 kJ/mol 3rd ionization energy is even higher than 2nd. Successive ionization energies increase in magnitude because # of e−s, which cause repulsion, steadily decrease. Not a smooth curve. Big jump in ionization energy after atom has lost valence electrons. Atom with same electronic conﬁguration as noble gas holds more strongly on to its e−s. Energy to remove e−s beyond valence electrons is signiﬁcantly greater than energy of chemical reactions and bonding. Only valence e−s (i.e., e−s outside of noble gas core) are involved in chemical reactions.

P. J. Grandinetti Chapter 21: Multi-electron Atoms 1st ionization energy Period Period Period Period Period 1 2 3 4 5 He Ne Ar Kr 2500

2000 F 1500 N Cl O H P 1000 Be C Mg S Si B 500 Al Li Na first ionization energy / kJ/mol 0 0 He Be N 10 Mg 20 30 40 50 atomic number 1st ionization energies generally increase as atomic radius decreases. Two clear behaviors are observed: 1st ionization energy decreases down a group. 1st ionization energy increases across period.

P. J. Grandinetti Chapter 21: Multi-electron Atoms 1st ionization energy

1st ionization energy decreases down a group as highest energy e−s are, on average, farther from nucleus. As n increases, size of orbital increases and e− is easier to remove. > > e.g., I1(Na) I1(Cs) and I1(Cl) I1(I).

He Ne 2500 1st ionization energy increases across period. 2000 ▶ e− with same n do not completely shield increasing nuclear F charge, so outermost e−s are held more tightly and require 1500 N > more energy to be ionized, e.g., I1(Cl) I1(Na) and O > 1000 Be C I1(S) I1(Mg). ▶ Plot of ionization energy versus Z is not perfect line but B 500 exceptions are easily explained: ﬁlled and half-ﬁlled subshells Li

have added stability just as ﬁlled shells show increased first ionization energy / kJ/mol > > 0 stability—e.g., this explains I1(Be) I1(B) and I1(N) I1(O). 0 10 atomic number

P. J. Grandinetti Chapter 21: Multi-electron Atoms Electron Aﬃnity

Energy associated with addition of electron to gaseous atom is called electron aﬃnity.

− → − . Cl(g) + e Cl (g) ΔE = −EEA = −349 kJ/mol

Sign of energy change is negative because energy is released in this process.

More negative ΔE corresponds to greater attraction for e−.

An unbound e− has an ΔE of zero.

− Convention is to deﬁne e aﬃnity energy, EEA, as negative of energy released in reaction.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Electron Aﬃnity Period Period Period Period Period 1 2 3 4 5 He Ne Ar Kr 350 Cl F I 300

250

200 C S 150 Si O 100 H Li Na electron affinity / kJ/mol 50 B P Al 0 He Be N 10 Mg 20 30 40 50 atomic number 2 rules that govern periodic trends of e− affinities: 1 e− affinity becomes smaller down group. 2 e− affinity decreases or increases across period depending on electronic configuration.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Electron Aﬃnity

1 e− aﬃnity becomes smaller down group.

▶ As n increases, size of orbital increases and aﬃnity for e− is less.

▶ Change is small and there are many exceptions.

2 e− aﬃnity decreases or increases across period depending on electronic conﬁguration.

▶ This occurs because of same sub-shell rule that governs ionization energies. > e.g., EEA(C) EEA(N). ▶ Since a half-ﬁlled p sub-shell is more stable, C has a greater aﬃnity for an electron than N.

▶ Obviously, the halogens, which are one electron away from a noble gas electron conﬁguration, have high aﬃnities for electrons.

▶ F’s electron aﬃnity is smaller than Cl’s because of higher electron–electron repulsions in smaller 2p orbital compared to larger 3p orbital of Cl.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Angular Momentum in Multi-electron Atoms

P. J. Grandinetti Chapter 21: Multi-electron Atoms Angular Momentum in Atom with N electrons In absence of external ﬁelds split ̂ for multi-electron atom into 3 parts: ̂ ̂ ̂ ̂ = 0 + ee + SO ̂ − 0 is electron kinetic energy and nuclear Coulomb attraction, i.e., non-interacting e s: [ ] ∑N ℏ2 Zq2 ̂ = − ∇2 − e 0 2m i 4휋휖 r i=1 e 0 i ̂ ee is electron repulsion: N∑−1 ∑N q2 ̂ = e ee 4휋휖 r i=1 j=i+1 0 ij ̂ SO is spin-orbit coupling: ∑N ̂ 휉 퓁⃗ ⋅ ⃗ SO = (ri) i si i=1 There are additional interactions we could add to total ̂ , but these can be handled later as perturbations to ̂ . P. J. Grandinetti Chapter 21: Multi-electron Atoms Angular Momentum in Atom with N electrons In absence of external ﬁelds split ̂ for multi-electron atom into 3 parts:

̂ ̂ ̂ ̂ = 0 + ee + SO

̂ In H-atom we saw that spin orbit coupling was small correction to 0. ̂ ̂ In He-atom with 2 electrons we also ﬁnd that SO is small compare to ee. < ̂ For multi-electron atoms with low number of electrons (N 40) we can take SO as small ̂ perturbation compared to ee, and take ̂ (0) ̂ ̂ . I = 0 + ee for low Z atoms

̂ 4 ≫ ̂ On other hand, we know that SO increases as Z . So when N 40 we ﬁnd that ee is small ̂ perturbation compared to SO, and take ̂ (0) ̂ ̂ . II = 0 + SO for high Z atoms

P. J. Grandinetti Chapter 21: Multi-electron Atoms Low Z multi-electron atoms

P. J. Grandinetti Chapter 21: Multi-electron Atoms Angular Momentum in Multi-electron Atom : Low Z case Total orbital and spin angular momenta, L⃗ and S⃗, respectively, are given by ∑N ∑N ⃗ 퓁⃗ ⃗ ⃗ L = i and S = si i=1 i=1 퓁⃗ ⃗ where i and si are individual electron orbital and spin angular momenta, respectively. The z components of L⃗ and S⃗ are given by sums ∑N ∑N ̂ 휓 ℏ휓 ̂ 휓 ℏ휓 Lz = ML where ML = m퓁,i and Sz = MS where MS = ms,i i=1 i=1

⃗̂ ⃗̂ ⃗̂ ̂ (0) ̂ (0), ⃗̂ The total angular momentum, J = L + S, commutes with I , i.e., [ I J] = 0. ̂ (0), ⃗̂ ̂ (0), ⃗̂ In low Z case we ﬁnd [ I L] = 0 and [ I S] = 0, thus can approximate total orbital and spin angular momenta as separately conserved—when spin-orbit coupling is negligible. This limit is called Russell-Saunders coupling or LS coupling. Using L, S and J we label energy levels using atomic term symbols.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Multi-electron Atom Term Symbols

Atomic spectroscopists have devised approach for specifying set of states having particular set of L, S, and J with a term symbol. Given L, S, and J for electron term symbol is deﬁned as

2S+1 {L symbol}J

2S + 1 is called spin multiplicity L symbol is assigned based on numerical value of L according to

L = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ← numerical value {L symbol} ≡ SPDFGHIKLMOQRT ← symbol

continuing afterwards in alphabetical order.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Multi-electron Atom Term Symbols

Example What term symbol is associated with the 1s2 conﬁguration?

electron 1 electron 2 total

m퓁 ms m퓁 ms ML MS 0 1 0 − 1 0 0 2 2

Only one value of ML = 0 in this conﬁguration implying that L = 0.

Likewise, only one value of MS = 0 implying that S = 0, (and 2S + 1 = 1)

Since J = L + S = 0 has only one MJ value of MJ = 0. For term symbol, 1s2 conﬁguration has S = 0, L = 0, and J = 0. 1 Thus, it belongs to term S0 (pronounced “singlet S zero”).

P. J. Grandinetti Chapter 21: Multi-electron Atoms Example What term symbol is associated with the 2p6 conﬁguration?

Solution: From orbital diagram we see that both MS and ML sum to zero,

1 Only J = 0 works so it belongs to S0 term. Note that we can ignore ﬁlled subshells. We get same result when considering all subshells.

1 All closed shells and subshells are S0 terms.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Example What are the terms associated with the 1s12s1 conﬁguration of helium?

Solution:. There are four possible states with 1s12s1.

All states have ML = 0, so we must have L = 0. , , highest MS value is +1 so need S = 1 which leads to MS = −1 0 +1.

leaves 1 state with MS = 0 and ML = 0, must belong to S = 0, and J = 0. 1s12s1 conﬁguration leads to 2 terms: 3 1 S1 term for L = 0, S = 1, J = 1 1 2 S0 term for L = 0, S = 0, J = 0

P. J. Grandinetti Chapter 21: Multi-electron Atoms Angular Momentum : Low Z case, Hund’s rules

When spin-orbit coupling is added back into Hamiltonian then degeneracy of diﬀerent J levels is removed.

We can follow Hund’s rules to determine which term corresponds to ground state:

1 Term with highest multiplicity, (2S + 1), is lowest in energy. Term energy increases with decreasing multiplicity.

2 For terms with same multiplicity, (2S + 1), term with highest L is lowest in energy.

3 If terms have same L and S, then for

1 less than half-ﬁlled sub-shells the term with smallest J is lowest in energy.

2 more than half-ﬁlled sub-shells the term with largest J is lowest in energy.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Example Find terms for 2p2 conﬁguration of C? Which term is associated with ground state?

Solution:( ) Number of ways to distribute n electrons to g spin orbitals in same sub-shell is G = G! = 6! = 15 ways for 2p2. N N!(G−N)! 2!(6−2)!

Largest ML magnitude is 2 with only MS = 0. Must be L = 2, 1 , , , , S = 0, J = 2, i.e., D2, with 5 states have ML = −2 −1 0 1 2. 1 With 5 states of D2 eliminated next largest ML magnitude remaining is 1 with MS values of −1, 0, and +1. Must have , , 3 3 3 L = 1, S = 1, J = 0 1 2, i.e., P0, P1, and P2. Corresponds to 9 states. With 14 states of 1D and 3P eliminated, only 1 state left. 1 Must be S = 0, L = 0, J = 0, i.e., S0. To determine ground state term follow Hund’s rule. ▶ 3 3 3 Find terms with highest multiplicity: P0, P1, and P2. ▶ Sub-shell is less than half-ﬁlled, so term with lowest J, 3 i.e., P0, is ground state.

P. J. Grandinetti Chapter 21: Multi-electron Atoms High Z multi-electron atoms

P. J. Grandinetti Chapter 21: Multi-electron Atoms Angular Momentum in Multi-electron Atom : High Z case

̂ (0) ̂ ̂ . II = 0 + SO for high Z atoms

⃗ Total angular momentum, Ĵ, is given by

∑N ⃗ ⃗, 퓁 J = ji where ji = i + si i=1

̂ (0), ⃗̂ ̂ (0),⃗̂ Find that [ II J] = 0 and [ II ji] = 0 ̂ (0), ⃗̂ ≠ ̂ (0), ⃗̂ ≠ But also ﬁnd that [ II L] 0 and [ II S] 0. ⃗ ⃗ In high Z limit orbital, L̂, and spin, Ŝ, angular momenta are not separately conserved. This is called the j-j coupling limit. Term symbols for labeling energy levels do not work in this case.

P. J. Grandinetti Chapter 21: Multi-electron Atoms Schrödinger Equation in Atomic Units

Potential V̂ (⃗r) for electron bound to positively charged nucleus is given by coulomb potential

2 −Zqe V(r) = 휋휖 4 0r

qe is fundamental unit of charge and Z is nuclear charge in multiples of qe. 3D Schrödinger equation for electron in hydrogen atom becomes [ ] ℏ Zq2 2 e 휓 ⃗ 휓 ⃗ − 휇 ∇ − 휋휖 (r) = E (r) 2 4 0r

In computational chemistry calculations it is easier to perform numerical calculations with all quantities expressed in atomic scale units to avoid computer roundoﬀ errors. Computational chemists go one step further and convert all distances in Schrödinger equation into . dimensionless quantities by dividing by Bohr radius, a0 = 52 9177210526763 pm

P. J. Grandinetti Chapter 21: Multi-electron Atoms Schrödinger Equation in Atomic Units Bohr radius is deﬁned as atomic unit of length. A dimensionless distance vector is deﬁned as

⃗′ ≡ ⃗ r r∕a0

Laplacian in terms of dimensionless quantities becomes ( ) ′ 휕2 휕2 휕2 휕2 휕2 휕2 ∇ 2 = a2∇2 = a2 + + = + + 0 0 휕x2 휕y2 휕z2 휕x′2 휕y′2 휕z′2

Converting wave function

dr⃗′ 1 |휓(⃗r)|2d⃗r = |휓(r⃗′)|2dr⃗′ then |휓(⃗r)|2 = |휓(r⃗′)|2 = |휓(r⃗′)|2 d⃗r a0

Then 1 electron Hamiltonian becomes [ ] 2 ℏ 1 ′ Zq 1 1 − ∇ 2 − e √ 휓(r⃗′) = E √ 휓(r⃗′) 2휇 2 4휋휖 a r′ a a a0 0 0 0 0

P. J. Grandinetti Chapter 21: Multi-electron Atoms Schrödinger Equation in Atomic Units ℏ2 q2 With atomic unit of energy, E = = e , and 휇 ≈ m assumption, Schrödinger Eq h 2 4휋휖 a e mea0 0 0 becomes [ ] ∇′2 Z E − − 휓(r⃗′) = E휓(r⃗′) h 2 r′ Dividing both sides by E and deﬁning dimensionless energy E′ = E∕E obtain h [ ] h ∇′2 Z − − 휓(r⃗′) = E′휓(r⃗′) 2 r′ Drop prime superscripts and write 1 e− atom dimensionless Schrödinger Eq as [ ] ∇2 Z − − 휓(⃗r) = E휓(⃗r) 2 r

Understood that all quantities are dimensionless and scaled by corresponding atomic unit quantity. To obtain “dimensionful” quantities corresponding dimensionless quantity must be multiplied by corresponding atomic unit. P. J. Grandinetti Chapter 21: Multi-electron Atoms Values of selected atomic units in SI base units. Quantity Unit Symbol Value in SI . −11 length a0 5 29177210526763 × 10 m . −34 mass me 9 10938356 × 10 kg . −19 charge qe 1 6021766208 × 10 C . −18 energy Eh 4 359744651391459 × 10 C angular momentum ℏ 1.054571800139113 × 10−34 J⋅s . −30 ⋅ electric dipole moment qea0 8 478353549661393 × 10 C m 2 2 . −41 2⋅ 2 electric polarizability qea0∕Eh 1 6488 × 10 C m ∕J . electric ﬁeld strength Eh∕(qea0) 5 14220671012897 V/m 3 . −30 3 probability density 1∕(a0) 6 748334508994266 × 10 /m The Hamiltonian for a multi-electron atom in atomic units is given by [ ] ∑N 2 N∑−1 ∑N ∇ Z 1 ̂ = − i − + . r r i=1 2 i i=1 j=i+1 ij

P. J. Grandinetti Chapter 21: Multi-electron Atoms