![Page 1: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/1.jpg)
Chapter 21Multi-electron Atoms
P. J. Grandinetti
Chem. 4300
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 2: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/2.jpg)
The Helium AtomTo a good approximation, we separate wave function into 2 parts: one for center of mass and other for2 electrons orbiting center of mass. Hamiltonian for 2 electrons of He-atom is
x
y
z
He = β β2
2meβ2
1
βββββKH (1)
β β2
2meβ2
2
βββββKH (2)
βZq2
e4ππ0r1
βββββVH (1)
βZq2
e4ππ0r2
βββββVH (2)
+q2
e4ππ0r12βββββ
Ve-e
1st and 2nd terms on left, KH(1) and KH(2), are kinetic energies for 2 electrons3rd and 4th terms, VH(1) and VH(2), are Coulomb attractive potential energy of each eβ to nucleus5th 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
![Page 3: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/3.jpg)
What happens if we ignore Ve-e, the eββeβ 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 eβs bound to He nucleus, so electron wave function is aproduct of single eβ wave functions:
πHe = πH(1)πH(2) Orbital approximation
Total energy using single eβ wave functions is
E = β(1 Ry)(
Z2
n21
+ Z2
n22
)
For both electrons in n1 = n2 = 1 states of He atom obtain E = -108.86 eVNeglecting repulsion gives βΌ 38% error compared to measured He ground state energy of β79.049 eV.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 4: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/4.jpg)
The Variational Method
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 5: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/5.jpg)
The Variational MethodMake a mathematical guess for ground state wave function, πguess. Is it a good guess?If we choose a well behaved and normalized wave function, πguess, then it can be expressed as a linearcombination of exact eigenstates, πn,
πguess =ββ
n=0cnπn
Exact eigenstates, πn, and coefficients, cn, needed to describe πguess are unknown.Since πguess is normalized we have
β«Vπβ
guessπguessdπ = 1 =ββ
n=0
ββm=0
cβmcn β«Vπβ
mπndπ
Exact eigenstates are orthonormal, so β«V πβmπndπ = 0 when m β n.
Thus we knowββ
n=0|cn|2 β«V
πβnπndπ =
ββn=0
|cn|2 = 1
Each cn is restricted to |cn| β€ 1.If πguess was perfect then πguess = π0 with c0 = 1 and cnβ 0 = 0
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 6: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/6.jpg)
The Variational MethodBack to guess and calculate expansion
β«Vπβ
guessπguessdπ =ββ
n=0
ββm=0
cβmcn β«Vπβ
mπndπ
Again, πn are exact eigenstates of , so β«V πβmπndπ = 0 when m β n.
β«Vπβ
guessπguessdπ =ββ
n=0|cn|2En β«V
πβnπndπ =
ββn=0
|cn|2En
Hope is that πguess β π0, with energy E0. Rearranging
β«Vπβ
guessπguessdπ =ββ
n=0|cn|2 (E0 + (En β E0)
)=
ββn=0
|cn|2E0+ββ
n=0|cn|2(EnβE0) = E0+
ββn=0
|cn|2(EnβE0)
Since both |cn|2 and (En β E0) can only be positive numbers we obtain
β«Vπβ
guessπguessdπ β₯ E0
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 7: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/7.jpg)
The Variational MethodThis is called the variational theorem.
β«Vπβ
guessπguessdπ β₯ E0
It tells us that any variation in guess that gives lower expectation value for β¨β© brings uscloser to correct ground state wave function and ground state energy.Suggests algorithm of
βΆ varying some parameters, a, b, c,β¦, in πguess(a, b, c,β¦ , x1, x2,β¦) andβΆ searching for minimum in
β«Vπβ
guess(a, b, c,β¦ , x1, x2,β¦)πguess(a, b, c,β¦ , x1, x2,β¦)dπ
βΆ to get best guess for πguess(abest, bbest, cbest,β¦ , x1, x2,β¦).
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 8: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/8.jpg)
ExampleApply variational method to trial wave function for He atom making orbital approximation,
πHe = πH(1)πH(2)
using πH(1) =1βπ
(Zβ²
a0
)3β2eβZβ²r1βa0 and πH(2) =
1βπ
(Zβ²
a0
)3β2eβZβ²r2βa0
and use Zβ² as variational parameter. Idea for replacing atomic number, Z, by Zβ² is that each electronsees smaller effective nuclear charge due to βscreeningβ by other electron.
Solution: Recalling
He = β β2
2meβ2
1βZq2
e4ππ0r1
β β2
2meβ2
2βZq2
e4ππ0r2
+q2
e4ππ0r12
β¨β© = β« πβH(1)
[β β2
2meβ2
1 βZq2
e
4ππ0r1
]βββββββββββββββββββββββββββββββ
H(1)
πH(1)dπ + β« πβH(2)
[β β2
2meβ2
2 βZq2
e
4ππ0r2
]βββββββββββββββββββββββββββββββ
H(2)
πH(2)dπ + β« πβH(1)π
βH(2)
[q2
e
4ππ0r12
]βββββββββββ
Ve-e
πH(1)πH(2)dπ
Define ΞZ = Z β Zβ² (or Z = Zβ² + ΞZ) ...P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 9: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/9.jpg)
Variational Method ExampleExpand and rearrange 1st and 2nd terms taking ΞZ = Z β Zβ² (or Z = Zβ² + ΞZ)
β¨β© = β« πβH(1)
[β β2
2meβ2
1 βZβ²q2
e
4ππ0r1
]πH(1)dπ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ1st
ββ« πβH(1)
ΞZq2e
4ππ0r1πH(1)dπ
βββββββββββββββββββββββββββββββββββββββ2nd
+ β« πβH(2)
[β β2
2meβ2
2 βZβ²q2
e
4ππ0r2
]πH(2)dπ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ3rd
β β« πβH(2)
ΞZq2e
4ππ0r2πH(2)dπ
βββββββββββββββββββββββββββββββββββ4th
+ β« πβH(1)π
βH(2)
[q2
e
4ππ0r12
]πH(1)πH(2)dπ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ5th
1st and 3rd terms are H-atom energies except with Zβ². We know these solutions.For 2nd and 4th terms we have
ββ« πβH(1)
ΞZq2e
4ππ0rπH(1)dπ = β
ΞZq2e
4ππ0
1π
(Zβ²
a0
)3
β«β
0rβ1eβ2Zβ²rβa04πr2dr = β
ΞZq2e
4ππ0
Zβ²
a0
5th term is more work and Iβll leave it as an exercise to show that
β« πβH(1)π
βH(2)
[q2
e4ππ0r12
]πH(1)πH(2)dπ =
5Zβ²q2e
4ππ0
18a0
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 10: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/10.jpg)
Variational Method Exampleβ¨β© = β2
[Zβ²2q2
e8ππ0a0
]βββββββββββββββ
1st, 3rd
β 2
[Zβ²ΞZq2
e4ππ0a0
]βββββββββββββββββββ
2nd,4th
+5Zβ²q2
e4ππ0
18a0
βββββββββββββββ5th
=[Zβ²2 β 2
(Z β 5
16
)Zβ²] q2
e4ππ0a0
Now comes minimization step where we calculate dβ¨β©βdZβ² = 0 as
dβ¨β©dZβ² = 2
[Zβ² β
(Z β 5
16
)Zβ²] q2
e
4ππ0a0= 0 which gives ΞZ = Z β Zβ² = 5β16.
Thus we find β¨β© = β(
Z β 516
)2 q2e
4ππ0a0β₯ E0
For He (Z = 2) we find Zβ² = Z β ΞZ = 2 β 5β16 = 1.6875, the effective nuclear charge after screeningby other eβ. For ground state energy this leads to
β¨β© = β(27
16
)2 q2e
4ππ0a0= β2.85
q2e
4ππ0a0= β77.5 eV,
error is only 1.9% compared to measured He ground state energy of β79.049 eV.P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 11: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/11.jpg)
Making the Orbital Approximation Good
The orbital approximation β the idea that electrons occupy individual orbitals β is notsomething chemists would give up easily.
In orbital approximation we write He atom ground state as
π = 1β2π1s(1)π1s(2) [πΌ(2)π½(1) β πΌ(1)π½(2)]
But now you know that electronβelectron repulsion term spoils this picture, making it only anapproximation.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 12: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/12.jpg)
Mean field potential energyIn 1927 Douglass Hartree proposed an approach for solving multi-electron SchrΓΆdinger Eq thatmakes orbital approximation reasonably good.In Hartreeβs approach each single electron is presumed to move in combined potential field of
nucleus, taken as point charge Zqe, andall other electrons, taken as continuous negative charge distribution.
Vi(r) β βZq2
e4πr
+Nβ
j=1,jβ i
q2e
4ππ0 β«V
|||πj(r)|||2|||r β rβ²||| d3rβ²
Vi(r) is βmean fieldβ potential energy for ith electron in multi-electron atom.
Details of Hartreeβs 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
![Page 13: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/13.jpg)
Douglas Hartree with Mathematician Phyllis Nicolson at the Hartree Differential Analyser atManchester University in 1935.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 14: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/14.jpg)
Making the Orbital Approximation GoodConsequence of Hartreeβs successful solution is that chemists think of atoms and molecules interms 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
Zeff = Z β π
Z is full nuclear charge, and π is shielding by average # of eβs between nucleus and ith eβ.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 15: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/15.jpg)
Making the Orbital Approximation Good
Zeff = Z β π
In multi-electron atom π increases with increasing π.
βΆ This is because electrons in s orbital have greater probability of being near nucleus than porbital, so electron of interest in s orbital sees greater fraction of nuclear charge than if it wasa p orbital.
βΆ Likewise when electron of interest is in p orbital it sees greater fraction of nuclear chargethan if it was in d orbital.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 16: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/16.jpg)
Energy of ith electron varies with both n and πEn
ergy
Distinguish electrons with same n and πvalue as being in same subshell.Ground state electron configuration isobtain by filling orbitals one electron at atime, in order of increasing energy startingwith lowest energy.Electrons in outer most shell of atom arevalence electrons, while electrons in innerclosed shells are core electrons.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 17: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/17.jpg)
Periodic Trends and TableBefore QM there were many attempts to arrange known elements so that there were somecorrelations between their known properties.
in 1869 Dimitri Mendeleev made first reasonably successful attempt by arranging elements in orderof increasing atomic mass, and, most importantly, found that elements with similar chemical andphysical properties occurred periodically. He placed these similar elements under each other incolumns.
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 ofincreasing 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
![Page 18: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/18.jpg)
1HHydrogen
1.008
3LiLithium
6.941
11NaSodium
22.990
19KPotassium
39.098
37RbRubidium
85.47
4BeBerylium
9.012
12MgMagnesium
24.305
20CaCalcium
40.08
38SrStrontium
87.62
55CsCesium
132.91
56BaBarium
137.33
87FrFrancium
(223)
21ScScandium
44.96
22TiTitanium
47.88
23VVanadium
50.94
39YYttrium
88.91
40ZrZirconium
91.22
57LaLanthanum
138.91
72HfHafnium
178.49
59PrPraseodymium
140.9077
73TaTantalum
180.95
41NbNiobium
92.91
74WTungsten
183.85
42MoMolybdenum
95.94
24CrChromium
52.00
60NdNeodymium
144.24
75ReRhenium
186.2
43TcTechnetium
(98)
25MnManganese
54.94
76OsOsmium
190.2
44RuRuthenium
101.1
26FeIron
55.85
27CoCobalt
58.93
28NiNickel
58.69
29CuCopper
63.546
30ZnZinc
65.38
45RhRhodium
102.91
46PdPalladium
106.4
47AgSilver
107.87
48CdCadmium
112.41
77IrIridium
192.2
78PtPlatinium
195.08
79AuGold
196.97
80HgMercury
200.59
5BBoron
10.81
6CCarbon
12.011
7NNitrogen
14.007
8OOxygen
15.999
9FFluorine
18.998
Neon
20.179
10Ne
13AlAluminum
26.98
14SiSilicon
28.09
15PPhosphorus
30.974
16SSulfur
32.06
17ClChlorine
35.453
18ArArgon
39.948
2HeHelium
4.003
31GaGallium
69.72
32GeGermanium
72.59
33AsArsenic
74.92
34SeSelenium
78.96
35BrBromine
79.904
36KrKrypton
83.80
49InIndium
114.82
50SnTin
118.69
51SbAntimony
121.75
52TeTellurium
127.60
53IIodine
126.90
54XeXenon
131.29
81TlThallium
204.38
82PbLead
207.2
83BiBismuth
208.98
84PoPolonium
(244)
85AtAstatine
(210)
86RnRadon
(222)
88RdRadium
226.03
89AcActinium
227.03
58CeCerium
140.12
61PmPromethium
(145)
62SmSamarium
150.36
90ThThorium
232.04
91PaProtactinium
231.0359
92UUranium
238.03
93NpNeptunium
237.05
63EuEuropium
151.96
64GdGadolinium
157.25
65TbTerbium
158.93
66DyDysprosium
162.50
67HoHolmium
164.93
68ErErbium
167.26
94PuPlutonium
(244)
95AmAmericium
(243)
96CmCurium
(247)
97BkBerkelium
(247)
98CfCalifornium
(251)
99EsEinsteinium
(254)
100FmFermium
(257)
101MdMendelevium
(258)
69TmThullium
168.93
70YbYtterbium
173.04
71LuLutetium
174.97
102NoNobellium
(259)
103LrLawrencium
(260)
IA
IIA
IIIB IVB VB VIB }VIIIBIB IIBVIIB
IIIA IVA VA VIA VIIA
VIIIA
1
2
3
4
5
6
7
Lanthanide Series
Actinide Series
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 19: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/19.jpg)
Atomic radii
100 20 30 40 50
50
0
100
150
200
250
atom
ic ra
dii/p
m
Period1
Period2
Period3
Period4
Period5
Be
Ne Kr
Mg
He
He N
Ar
1 Size generally decreases across period from left to right because outermost valence electrons aremore strongly attracted to the nucleus.
βΆ Moving across period inner electrons ability to cancel increasing nuclear charge diminishesβΆ Inner electrons in s orbitals shield nuclear charge better than those in p orbitals.βΆ 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 meanslarger orbitals and an increase in atom size.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 20: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/20.jpg)
Ionization EnergyEnergy required to remove an eβ from gaseous atom or ion is called ionization energy.
First ionization energy, I1, is energy required to remove highest energy eβ from neutralgaseous atom.
Na(g) β Na+(g) + eβ I1 = 496 kJ/mol.
Ionization energy is positive because it requires energy to remove eβ.Second ionization energy, I2, is energy required to remove 2nd eβ from singly chargedgaseous cation.
Na+(g) β Na2+(g) + eβ I2 = 4560 kJ/mol.
Second ionization energy is almost 10 times that of 1st because number of eβs causingrepulsions is reduced.Third ionization energy, I3, is energy required to remove 3rd electron from doubly chargedgaseous cation.
Na2+(g) β Na3+(g) + eβ I3 = 6913 kJ/mol.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 21: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/21.jpg)
Ionization Energy
Na(g) β Na+(g) + eβ I1 = 496 kJ/mol.
Na+(g) β Na2+(g) + eβ I2 = 4560 kJ/mol.
Na2+(g) β Na3+(g) + eβ I3 = 6913 kJ/mol.
3rd ionization energy is even higher than 2nd.Successive ionization energies increase in magnitude because # of eβs, which causerepulsion, 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 eβs.Energy to remove eβs beyond valence electrons is significantly greater than energy ofchemical reactions and bonding.Only valence eβs (i.e., eβs outside of noble gas core) are involved in chemical reactions.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 22: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/22.jpg)
1st ionization energy
100 20 30 40 50
500
0
1000
1500
2000
2500
atomic number
first
ioni
zatio
n en
ergy
/ kJ
/mol
Period1
Period2
Period3
Period4
Period5
Li
Be
Be
B
CO
F
Cl
Ne Kr
Na
Mg
Mg
Al
Si
PS
He
He
H
N
N
Ar
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
![Page 23: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/23.jpg)
1st ionization energy
1st ionization energy decreases down a group as highest energy eβs are, on average, farther fromnucleus. As n increases, size of orbital increases and eβ is easier to remove.e.g., I1(Na) > I1(Cs) and I1(Cl) > I1(I).
1st ionization energy increases across period.βΆ eβ with same n do not completely shield increasing nuclear
charge, so outermost eβs are held more tightly and requiremore energy to be ionized, e.g., I1(Cl) > I1(Na) andI1(S) > I1(Mg).
βΆ Plot of ionization energy versus Z is not perfect line butexceptions are easily explained: filled and half-filled subshellshave added stability just as filled shells show increasedstabilityβe.g., this explains I1(Be) > I1(B) and I1(N) > I1(O). 100
500
0
1000
1500
2000
2500
atomic number
first
ioni
zatio
n en
ergy
/ kJ
/mol
Li
Be
B
CO
F
NeHe
N
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 24: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/24.jpg)
Electron Affinity
Energy associated with addition of electron to gaseous atom is called electron affinity.
Cl(g) + eβ β Clβ(g) ΞE = βEEA = β349 kJ/mol.
Sign of energy change is negative because energy is released in this process.
More negative ΞE corresponds to greater attraction for eβ.
An unbound eβ has an ΞE of zero.
Convention is to define eβ affinity energy, EEA, as negative of energy released in reaction.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 25: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/25.jpg)
Electron Affinity
10 20 30 40 50
50
0
100
150
200
250
300
350
atomic number
elec
tron
affin
ity /
kJ/m
ol
Period1
Period2
Period3
Period4
Period5
Li
Be
B
C
O
FCl
I
Ne Kr
Na
Mg
Al
Si
P
S
He
He
H
N
Ar
2 rules that govern periodic trends of eβ affinities:1 eβ affinity becomes smaller down group.2 eβ affinity decreases or increases across period depending on electronic configuration.P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 26: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/26.jpg)
Electron Affinity1 eβ affinity becomes smaller down group.
βΆ As n increases, size of orbital increases and affinity for eβ is less.
βΆ Change is small and there are many exceptions.
2 eβ affinity decreases or increases across period depending on electronic configuration.
βΆ This occurs because of same sub-shell rule that governs ionization energies.e.g., EEA(C) > EEA(N).
βΆ Since a half-filled p sub-shell is more stable, C has a greater affinity for an electron than N.
βΆ Obviously, the halogens, which are one electron away from a noble gas electronconfiguration, have high affinities for electrons.
βΆ Fβs electron affinity is smaller than Clβs because of higher electronβelectron repulsions insmaller 2p orbital compared to larger 3p orbital of Cl.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 27: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/27.jpg)
Angular Momentum in Multi-electron Atoms
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 28: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/28.jpg)
Angular Momentum in Atom with N electronsIn 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 eβs:
0 =Nβ
i=1
[β β2
2meβ2
i βZq2
e4ππ0ri
]ee is electron repulsion:
ee =Nβ1βi=1
Nβj=i+1
q2e
4ππ0rij
SO is spin-orbit coupling:
SO =Nβ
i=1π(ri)πi β si
There are additional interactions we could add to total , but these can be handled later asperturbations to .
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 29: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/29.jpg)
Angular Momentum in Atom with N electronsIn 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 smallperturbation compared to ee, and take
(0)I = 0 + ee for low Z atoms.
On other hand, we know that SO increases as Z4. So when N β« 40 we find that ee is smallperturbation compared to SO, and take
(0)II = 0 + SO for high Z atoms.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 30: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/30.jpg)
Low Z multi-electron atoms
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 31: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/31.jpg)
Angular Momentum in Multi-electron Atom : Low Z caseTotal orbital and spin angular momenta, L and S, respectively, are given by
L =Nβ
i=1πi and S =
Nβi=1
si
where πi and si are individual electron orbital and spin angular momenta, respectively.
The z components of L and S are given by sums
Lzπ = MLβπ where ML =Nβ
i=1mπ,i and Szπ = MSβπ where MS =
Nβi=1
ms,i
The total angular momentum, J = L + S, commutes with (0)I , i.e., [(0)
I , J] = 0.
In low Z case we find [(0)I , L] = 0 and [(0)
I , S] = 0, thus can approximate total orbital and spinangular momenta as separately conservedβwhen 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
![Page 32: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/32.jpg)
Multi-electron Atom Term Symbols
Atomic spectroscopists have devised approach for specifying set of states having particular setof 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 multiplicityL 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 β numerical value{L symbol} β‘ S P D F G H I K L M O Q R T β symbol
continuing afterwards in alphabetical order.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 33: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/33.jpg)
Multi-electron Atom Term Symbols
ExampleWhat term symbol is associated with the 1s2 configuration?
electron 1 electron 2 totalmπ ms mπ ms ML MS
0 12 0 β1
2 0 0
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.Thus, it belongs to term 1S0 (pronounced βsinglet S zeroβ).
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 34: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/34.jpg)
ExampleWhat term symbol is associated with the 2p6 configuration?
Solution: From orbital diagram we see that both MS and ML sum to zero,
Only J = 0 works so it belongs to 1S0 term.Note that we can ignore filled subshells.We get same result when considering all subshells.
All closed shells and subshells are 1S0 terms.P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 35: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/35.jpg)
ExampleWhat 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 = β1, 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:
1 3S1 term for L = 0, S = 1, J = 12 1S0 term for L = 0, S = 0, J = 0
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 36: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/36.jpg)
Angular Momentum : Low Z case, Hundβs rules
When spin-orbit coupling is added back into Hamiltonian then degeneracy of different J levelsis removed.
We can follow Hundβs rules to determine which term corresponds to ground state:
1 Term with highest multiplicity, (2S + 1), is lowest in energy. Term energy increases withdecreasing 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
![Page 37: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/37.jpg)
ExampleFind 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(GN
)= G!
N!(GβN)!= 6!
2!(6β2)!= 15 ways for 2p2.
Largest ML magnitude is 2 with only MS = 0. Must be L = 2,S = 0, J = 2, i.e., 1D2, with 5 states have ML = β2,β1, 0, 1, 2.With 5 states of 1D2 eliminated next largest ML magnituderemaining is 1 with MS values of β1, 0, and +1. Must haveL = 1, S = 1, J = 0, 1, 2, i.e., 3P0, 3P1, and 3P2. Correspondsto 9 states.With 14 states of 1D and 3P eliminated, only 1 state left.Must be S = 0, L = 0, J = 0, i.e., 1S0.
To determine ground state term follow Hundβs rule.βΆ Find terms with highest multiplicity: 3P0, 3P1, and 3P2.βΆ Sub-shell is less than half-filled, so term with lowest J,
i.e., 3P0, is ground state.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 38: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/38.jpg)
High Z multi-electron atoms
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 39: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/39.jpg)
Angular Momentum in Multi-electron Atom : High Z case
(0)II = 0 + SO for high Z atoms.
Total angular momentum, J, is given by
J =Nβ
i=1ji, where ji = πi + si
Find that [(0)II ,
J] = 0 and [(0)II ,ji] = 0
But also find that [(0)II ,
L] β 0 and [(0)II ,
S] β 0.
In high Z limit orbital, L, and spin, S, 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
![Page 40: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/40.jpg)
SchrΓΆdinger Equation in Atomic Units
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 41: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/41.jpg)
SchrΓΆdinger Equation in Atomic UnitsPotential V(r) for electron bound to positively charged nucleus is given by coulomb potential
V(r) =βZq2
e4ππ0r
qe is fundamental unit of charge and Z is nuclear charge in multiples of qe.
3D SchrΓΆdinger equation for electron in hydrogen atom becomes[β β
2πβ2 β
Zq2e
4ππ0r
]π(r) = Eπ(r)
In computational chemistry calculations it is easier to perform numerical calculations with all quantitiesexpressed in atomic scale units to avoid computer roundoff errors.
Computational chemists go one step further and convert all distances in SchrΓΆdinger equation intodimensionless quantities by dividing by Bohr radius, a0 = 52.9177210526763 pm
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 42: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/42.jpg)
SchrΓΆdinger Equation in Atomic UnitsBohr radius is defined as atomic unit of length. A dimensionless distance vector is defined as
rβ² β‘ rβa0
Laplacian in terms of dimensionless quantities becomes
ββ²2 = a2
0β2 = a2
0
(π2
πx2 + π2
πy2 + π2
πz2
)= π2
πxβ²2+ π2
πyβ²2+ π2
πzβ²2
Converting wave function
|π(r)|2dr = |π(rβ²)|2drβ² then |π(r)|2 = |π(rβ²)|2 drβ²
dr= 1
a0|π(rβ²)|2
Then 1 electron Hamiltonian becomes[β β
2π1a2
0
ββ²2 β
Zq2e
4ππ0a0rβ²
]1βa0π(rβ²) = E 1β
a0π(rβ²)
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 43: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/43.jpg)
SchrΓΆdinger Equation in Atomic UnitsWith atomic unit of energy, Eh = β2
mea20
=q2
e4ππ0a0
, and π β me assumption, SchrΓΆdinger Eq
becomesEh
[βββ²2
2β Z
rβ²
]π(rβ²) = Eπ(rβ²)
Dividing both sides by Eh and defining dimensionless energy Eβ² = EβEh obtain[βββ²2
2β Z
rβ²
]π(rβ²) = Eβ²π(rβ²)
Drop prime superscripts and write 1 eβ atom dimensionless SchrΓΆdinger Eq as[ββ2
2β Z
r
]π(r) = Eπ(r)
Understood that all quantities are dimensionless and scaled by corresponding atomic unit quantity.To obtain βdimensionfulβ quantities corresponding dimensionless quantity must be multiplied bycorresponding atomic unit.
P. J. Grandinetti Chapter 21: Multi-electron Atoms
![Page 44: Chapter 21 - Multi-electron AtomsIn orbital approximation we write He atom ground state as π= 1 β 2 π1s(1)π1s(2)[πΌ(2)π½(1)βπΌ(1)π½(2)] But now you know that electronβelectron](https://reader036.vdocuments.site/reader036/viewer/2022062603/5f0395897e708231d409c5a1/html5/thumbnails/44.jpg)
Values of selected atomic units in SI base units.Quantity Unit Symbol Value in SIlength a0 5.29177210526763 Γ 10β11 mmass me 9.10938356 Γ 10β34 kg
charge qe 1.6021766208 Γ 10β19 Cenergy Eh 4.359744651391459 Γ 10β18 C
angular momentum β 1.054571800139113 Γ 10β34 Jβ selectric dipole moment qea0 8.478353549661393 Γ 10β30 Cβ melectric polarizability q2
ea20βEh 1.6488 Γ 10β41 C2β m2βJ
electric field strength Ehβ(qea0) 5.14220671012897 V/mprobability density 1β(a3
0) 6.748334508994266 Γ 10β30 /m3
The Hamiltonian for a multi-electron atom in atomic units is given by
=Nβ
i=1
[ββ2
i2
β Zri
]+
Nβ1βi=1
Nβj=i+1
1rij.
P. J. Grandinetti Chapter 21: Multi-electron Atoms