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Chapter 18Electronic Structure of the Hydrogen Atom
P. J. Grandinetti
Chem. 4300
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Electronic Structure of the Hydrogen Atom
Hydrogen atom is simplest atomic system where Schrödinger equation can be solvedanalytically and compared to experimental measurements.
Analytical solution serve as basis for obtaining approximate solutions for multi-electronatoms and molecules, where no analytical solution exists.
Warning: Working through analytical solution of H-atom may cause drowsiness. Do notoperate heavy machinery during this derivation.
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Two Particle Problem
x
y
z2 particles with mass mp and me at r⃗p and r⃗e.Total mass, center of mass, and inter-particle distance vector is
M = mp + me, R⃗ =mpr⃗p + mer⃗e
M, and r⃗ = r⃗e − r⃗p
Express r⃗p and r⃗e in terms of M, R⃗ and r⃗ as
r⃗p = R⃗ −meM
r⃗ and r⃗e = R⃗ +mp
Mr⃗
Express individual momenta of 2 particles as
p⃗p = mpdr⃗p
dtand p⃗e = me
dr⃗edt
then in terms of M, R⃗ and r⃗ as
p⃗p = mp
(dR⃗dt
−meM
dr⃗dt
)and p⃗e = me
(dR⃗dt
+mp
Mdr⃗dt
)P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Two Particle ProblemNext, consider total energy
E =p⃗p
2
2mp+
p⃗e2
2me+ V(r⃗)
V(r⃗) depends only on distance between 2 particles.Write energy in terms of M, R⃗ and r⃗, and obtain
E = 12
M(
dR⃗dt
)2
+ 12𝜇(
dr⃗dt
)2+ V(r⃗)
𝜇 is reduced mass, given by1𝜇= 1
mp+ 1
me
Define 2 new momenta associated with center of mass and reduced mass,
p⃗R = M dR⃗dt, and p⃗r = 𝜇dr⃗
dtP. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Two Particle Problem
Total energy becomes
E =p⃗R
2
2M+
p⃗r2
2𝜇+ V(r⃗)
1st term is translational energy of center of mass
2nd term is kinetic energy due to relative motion of 2 particles
Translating to quantum mechanics, we writetime independent Schrödinger equation for 2 particle system as[
p⃗R2
2M+
p⃗r2
2𝜇+ V(r⃗)
]f (R⃗, r⃗) =
[− ℏ2
2M∇⃗2
R−ℏ2
2𝜇∇⃗2
r + V(r⃗)]
f (R⃗, r⃗) = Ef (R⃗, r⃗)
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Two Particle Problem[− ℏ2
2M∇⃗2
R − ℏ2
2𝜇∇⃗2
r + V(r⃗)]
f (R⃗, r⃗) = Ef (R⃗, r⃗)
Wave function can be separated into product of two wave functions
f (R⃗, r⃗) = 𝜒(R⃗)𝜓(r⃗)
𝜒(R⃗) depending only on center of mass𝜓(r⃗) depending only on relative motion of 2 particlesSubstitute product into Schrödinger Eq above, we obtain 2 wave equations
− ℏ2
2M∇⃗2
R𝜒(R⃗) = ER𝜒(R⃗) and[−ℏ
2
2𝜇∇⃗2
r + V(r⃗)]𝜓(r⃗) = Er𝜓(r⃗)
whereE = ER + Er
On left is wave equation for translational motion of free particle of mass MOn right is wave equation for particle with mass 𝜇 in potential V(r⃗)For electron bound positively charged nucleus, we focus on PDE for 𝜓(r⃗)
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Schrödinger Equation in Spherical Coordinates
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Schrödinger Equation in Spherical Coordinates
Focus on PDE for 𝜓(r⃗): Electron bound to positively charged nucleus[−ℏ
2
2𝜇∇2 + V̂(r)
]𝜓(r⃗) = E𝜓(r⃗)
Coulomb potential is
V̂(r) = −Zq2
e4𝜋𝜖0r
Central potential with 1∕rdependence.V → −∞ as r → 0V → 0 as r → ∞
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Schrödinger Equation in Spherical CoordinatesSince V̂(r) only depends on r, we adopt spherical coordinates
−ℏ2
2𝜇
[𝜕2
𝜕r2 + 2r𝜕𝜕r
+ 1r2
(1
sin 𝜃𝜕𝜕𝜃
(sin 𝜃 𝜕
𝜕𝜃
)+ 1
sin2 𝜃𝜕2
𝜕𝜙2
)⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
−L̂2∕ℏ2
]
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟∇2
𝜓 + V̂(r)𝜓 = E𝜓
Take term in parentheses as −L̂2∕ℏ2, rearrange, and simplify to
r2 𝜕2𝜓(r, 𝜃, 𝜙)𝜕r2 + 2r
𝜕𝜓(r, 𝜃, 𝜙)𝜕r
+2𝜇r2
ℏ2
(E − V̂(r)
)𝜓(r, 𝜃, 𝜙) = 1
ℏ2 L̂2𝜓(r, 𝜃, 𝜙)
To solve PDE use separation of variables
𝜓(r, 𝜃, 𝜙) = R(r)Y(𝜃, 𝜙)
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Schrödinger Equation in Spherical CoordinatesSubstitute 𝜓(r, 𝜃, 𝜙) into PDE, and dividing both sides by 𝜓(r, 𝜃, 𝜙)
r2
R(r)𝜕2R(r)𝜕r2 + 2r
R(r)𝜕R(r)𝜕r
+2𝜇r2
ℏ2
(E − V̂(r)
)= 1
Y(𝜃, 𝜙)1ℏ2 L̂2Y(𝜃, 𝜙)
Left side depends only on rRight side depends only on 𝜃 and 𝜙.We have turned one PDE into two ODEs.
1 We know eigenfunctions and eigenvalues of L̂2 on right side,
L̂2Y𝓁,m(𝜃, 𝜙) = 𝓁(𝓁 + 1)ℏ2Y𝓁,m(𝜃, 𝜙)
2 Define 𝓁(𝓁 + 1) as separation constant and obtain ODE for radial part
r2
R(r)d2R(r)
dr2 + 2rR(r)
dR(r)dr
+2𝜇r2
ℏ2
(E − V̂(r)
)= 𝓁(𝓁 + 1)
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Schrödinger Equation in Spherical CoordinatesExpanding and rearranging ODE for radial part, we obtain[
−ℏ2
2𝜇
(d2
𝜕r2 + 2r
ddr
)+ ℏ2
2𝜇𝓁(𝓁 + 1)
r2 + V̂(r)⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
Veff(r)
]R(r) = ER(r)
Looks like 1D time independent Schrödinger Eq with effective potential
V̂eff(r) =ℏ2
2𝜇𝓁(𝓁 + 1)
r2⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟
Centrifugal Term
−Zq2
e4𝜋𝜖0r
Angular momentum,√𝓁(𝓁 + 1)ℏ, creates centrifugal force that pushes electron away from
nucleus.Notice 2 terms in V̂eff(r) always have opposite signs.
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Plot of Veff for 𝓁 = 0, 1, 2, and 3.
20 4 6 8 10 12 14-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
Note: potential minimum shifts to higher radii with increasing angular momentum (i.e., centrifugalforce).
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
V̂eff(r): Effective Potential of H-Like AtomSwitching to Atomic Units: Convenient units for atomic physics
Atomic unit of length, also called Bohr radius, is defined as
a0 ≡ 4𝜋𝜖0ℏ2
q2eme
= 52.9177210526763 pm
Atomic unit of energy is
1Eh ≡ ℏ2
mea20
=q2
e4𝜋𝜖0a0
= 27.21138602818051 eV
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Wave Equation for the Radial Part
[−ℏ
2
2𝜇
(d2
𝜕r2 + 2r
ddr
)+ ℏ2
2𝜇𝓁(𝓁 + 1)
r2 −Zq2
e4𝜋𝜖0r
]R(r) = ER(r)
pour yourself a fifth cup of coffee...
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Radial Wave EquationTo obtain radial wave function, R(r), we must solve[
−ℏ2
2𝜇
(d2
𝜕r2 + 2r
ddr
)+ ℏ2
2𝜇𝓁(𝓁 + 1)
r2 −Zq2
e4𝜋𝜖0r
]R(r) = ER(r)
Begin by dividing and both sides by E, and obtain
− ℏ2
2𝜇E
[d2
𝜕r2 + 2r
ddr
]R(r) +
(ℏ2
2𝜇E𝓁(𝓁 + 1)
r2 −Zq2
e4𝜋𝜖0rE
− 1
)R(r) = 0
Define𝜅2 ≡ −
2𝜇Eℏ2 or E = −ℏ
2𝜅2
2𝜇𝜅 is in wave numbers. Rearrange potential energy expression to
−Zq2
e4𝜋𝜖0rE
=Zq2
e4𝜋𝜖0r
2𝜇ℏ2𝜅2 =
Zq2e𝜇
2𝜋𝜖0ℏ2𝜅1𝜅r
=2𝜌0𝜅r
𝜌0 is a dimensionless quantity: 𝜌0 ≡ Zq2e𝜇
2𝜋𝜖0ℏ2𝜅P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Radial Wave EquationSubstitute expression for potential energy into ODE
1𝜅2
[d2
𝜕r2 + 2r
ddr
]R(r) −
(1 + 𝓁(𝓁 + 1)
(𝜅r)2+
2𝜌0𝜅r
)R(r) = 0
Define dimensionless variable: 𝜌 ≡ 2𝜅rRewrite radial part as function of 𝜌 as[
d2
d𝜌2 + 2𝜌
dd𝜌
]R(𝜌) −
(1 + 𝓁(𝓁 + 1)
𝜌2 +2𝜌0𝜌
)R(𝜌) = 0
Further simplify ODE by defining
u(𝜌) = 𝜌R(𝜌),du(𝜌)
d𝜌= 𝜌
dR(𝜌)d𝜌
+ R(𝜌),d2u(𝜌)
d𝜌2 = 𝜌d2R(𝜌)
d𝜌2 + 2dR(𝜌)
d𝜌
Re-express ODE asd2u(𝜌)
d𝜌2 −(
14−𝜌02𝜌
+ 𝓁(𝓁 + 1)𝜌2
)u(𝜌) = 0
𝜌 varies from 0 to ∞. 1st look at asymptotic solutions: (1) 𝜌→ ∞ and (2) 𝜌→ 0P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Radial Wave Equation, 𝜌→ ∞
Starting withd2u(𝜌)
d𝜌2 −(
14−𝜌02𝜌
+ 𝓁(𝓁 + 1)𝜌2
)u(𝜌) = 0
(1) 𝜌 → ∞. At large values of 𝜌 approximate ODE as
d2u(𝜌)d𝜌2 −
u(𝜌)4
≈ 0
Solutions to this ODE have formu(𝜌) ∼ Ae−𝜌∕2 + Be𝜌∕2
Reject positive exponent since require that solution be finite everywhere,
u(𝜌) ∼ Ae−𝜌∕2 in limit that 𝜌 → ∞
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Radial Wave Equation, 𝜌→ 0Starting with
d2u(𝜌)d𝜌2 −
(14−𝜌02𝜌
+ 𝓁(𝓁 + 1)𝜌2
)u(𝜌) = 0.
(2) 𝜌 → 0. At small values of 𝜌 approximate ODE as
d2u(𝜌)d𝜌2 − 𝓁(𝓁 + 1)
𝜌2 u(𝜌) ≈ 0.
Solutions to this ODE have formu(𝜌) ∼ A𝜌𝓁+1 + B𝜌−𝓁 .
Reject B term again because require that solution be finite at 𝜌 = 0.
u(𝜌) ∼ A𝜌𝓁+1 in limit that 𝜌→ 0
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Radial Wave Equation
u(𝜌) ∼ Ae−𝜌∕2 in limit that 𝜌 → ∞
u(𝜌) ∼ A𝜌𝓁+1 in limit that 𝜌 → 0
Now, back tod2u(𝜌)
d𝜌2 −(
14−𝜌02𝜌
+ 𝓁(𝓁 + 1)𝜌2
)u(𝜌) = 0
we propose general solution of form
u(𝜌) = 𝜌𝓁+1 L(𝜌) e−𝜌∕2
This has correct behavior in two limits.
Only need to determine L(𝜌) to get behavior for all 𝜌.
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Radial Wave EquationPlug u(𝜌) = 𝜌𝓁+1 L(𝜌) e−𝜌∕2, where 𝜌 ≡ 2𝜅r, into
d2u(𝜌)d𝜌2 −
(14−𝜌02𝜌
+ 𝓁(𝓁 + 1)𝜌2
)u(𝜌) = 0
gives
𝜌d2L(𝜌)
d𝜌2 +(
2𝓁 + 2⏟⏟⏟
k+1
−𝜌)dL(𝜌)
d𝜌+(𝜌0
2− (𝓁 + 1)
)⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟
j
L(𝜌) = 0
Set j = 𝜌0∕2 − (𝓁 + 1), and k = 2𝓁 + 1 and this ODE is recognized as
𝜌d2Lk
j (𝜌)
d𝜌2 + (k + 1 − 𝜌)dLk
j (𝜌)
d𝜌+ jLk
j (𝜌) = 0, associated Laguerre differential equation
Has nonsingular solutions only if j is non-negative integers, j = 0, 1, 2,…These solutions, Lk
j (𝜌), are called the associated Laguerre polynomials.P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Selected associated Laguerre polynomialsn 𝓁 L2𝓁+1
n−𝓁−1(𝜌) Polynomial Roots1 0 L1
0(𝜌) 1 -2 0 L1
1(𝜌) 2 − 𝜌 2 - -2 1 L3
0(𝜌) 1 - -3 0 L1
2(𝜌)12
(6 − 6𝜌 + 𝜌2) 1.26795 4.73205 -
3 1 L31(𝜌) 4 − 𝜌 4 - -
3 2 L50(𝜌) 1 - - -
4 0 L13(𝜌)
16
(24 − 36𝜌 + 12𝜌2 − 𝜌3) 7.75877 0.935822 3.30541
4 1 L32(𝜌)
12
(20 − 10𝜌 + 𝜌2) 2.76393 7.23607 -
4 2 L51(𝜌) 6 − 𝜌 6 - -
4 3 L70(𝜌) 1 - - -
5 0 L14(𝜌)
124
(120 − 240𝜌 + 120𝜌2 − 20𝜌3 + 𝜌4) 0.743292 2.57164 5.73118 10.9539
5 1 L33(𝜌)
16
(120 − 90𝜌 + 18𝜌2 − 𝜌3) 2.14122 5.31552 10.5433 -
5 2 L52(𝜌)
12
(42 − 14𝜌 + 𝜌2) 4.35425 9.64575 - -
5 3 L71(𝜌) 8 − 𝜌 8 - - -
5 4 L90(𝜌) 1 - - - -
Associated Laguerre polynomials following definition where Lkj (x) = (−1)k dk
dxk Lj+k(x).
Laguerre polynomial Lj(x) is defined by Rodrigues formula: Lj(x) =1n!
ex dj
dxj
(xje−x).
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Radial Wave EquationSince 𝓁 = 0, 1, 2,… , and j = 0, 1, 2,… , then
𝜌02
= j + 𝓁 + 1
can only take on integer values of 𝜌0∕2 = 1, 2,….We define this as the principal quantum number
n ≡ 𝜌0∕2 = j + 𝓁 + 1,
which can only take on values of
n = 1, 2, 3,…Wave functions with same n value form set called a shell.Special letters are sometimes assigned to each n value
n = 1 2 3 4 5 ← numerical valueK L M N O ← symbol
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Radial Wave Equation𝓁 is called the azimuthal quantum numberRearranging n = j + 𝓁 + 1 for 𝓁 gives
𝓁 = n − j − 1
and we find that 𝓁 cannot exceed n − 1 (since lowest value of j is zero).Range of 𝓁 is
𝓁 = 0,… , n − 1Recall that azimuthal quantum number, 𝓁, defines total angular momentum of
√𝓁(𝓁 + 1)ℏ.
m𝓁 is called the magnetic quantum number.m𝓁 has positive and negative integer values between −𝓁 and 𝓁.When x, y, or z component of the electron’s angular momentum is measured, only values ofm𝓁ℏ are observed.
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Solving the Radial Wave EquationWave functions with same value of n and 𝓁 form set called a sub-shell.Special letters are assigned to sub-shell with given 𝓁 value,
𝓁 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ← numerical values p d f g h i k l m o q r t ← symbol
and continue afterwards in alphabetical order.1st four letters arose in pre-quantum atomic spectroscopy for classifying emission lines and standfor sharp, principal, diffuse, and fine.Shorthand n𝓁 notation uses principal quantum number with 𝓁 symbol,Wave function with
▶ n = 1, 𝓁 = 0 is referred to as 1s state▶ n = 2, 𝓁 = 1 is referred to as 2p state
Number of roots of L2𝓁+1n−𝓁−1 is n − 𝓁 − 1. Thus, number of radial nodes is equal to n − 𝓁 − 1.
Recall for Spherical Harmonics that number of angular nodes is 𝓁
Thus, total number of nodes is (n − 𝓁 − 1) + 𝓁 = n − 1.P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Energy of the Hydrogen-Like AtomGiven the constraint that 𝜌0∕2 = n we go back to
𝜌0 ≡ Zq2e𝜇
2𝜋𝜖0ℏ2𝜅= 2n and rearrange to 𝜅n =
Zq2e𝜇
4𝜋𝜖0ℏ2n
From
𝜅2 ≡ −2𝜇Eℏ2 rearranges to En = −
ℏ2𝜅2n
2𝜇= −
Z2q4e𝜇
32𝜋2𝜖20ℏ
2n2= −
Z2q4e𝜇
8𝜖20h2n2
En = −Z2q4
e𝜇
8𝜖20h2n2
Energy of H-like atom
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Energy of the Hydrogen-Like Atom
En = −Z2q4
e𝜇
8𝜖20h2n2
Energy of H-like atom
e− bound to nucleus with charge Z energyonly depends on n and ZIf Z increases (holding n constant) thenEn decreases—becomes more negative.Higher Z means nucleus holds e− moretightlyIn limit that n → ∞ then E → 0 and e− isunbound
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Energy of the Hydrogen-Like AtomWith 𝜇 ≈ me, the n = 1 energy is called Rydberg unit of energy (Ry)
1Ry = −E1 =q4
eme
8𝜖20h2
= 13.60569301218355 eV
With 𝜇 ≈ me approximation
En = −1 Ry
n2 Energy of H atom
Can also divide by atomic unit of energy, Eh, to obtain
En = −1 Eh
2n2 Energy of H atom
Given n, energy is identical for each 𝓁. Given 𝓁 energy is identical for 2𝓁 + 1 values of m𝓁.
Degeneracy of nth energy level is gn =n−1∑𝓁=0
(2𝓁 + 1) = n + 2n−1∑𝓁=0
𝓁 = n + n(n − 1) = n2
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Energy levels of single electron bound to protonorbitals
UV light emission“Lyman series
Visible light emissionBalmer series”
degeneracy
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Energy of the Hydrogen-Like AtomRemember Balmer series for hydrogen emission spectra obeys relation:
1𝜆= RH
( 122 − 1
n2
)where RH = 1.097 × 107 /m
Followed by Bohr’s theory of atom which gave
1𝜆=(
14𝜋𝜖0
)2(
meq4e
4𝜋ℏ3c0
)Z2
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟perfect agreement with RH
(1n2
f
− 1n2
i
)
Taking h𝜈 = h𝜈𝜆= ΔE = Ef − Ei and solving for 1∕𝜆 from H-like atom energy gives exact
same result as Bohr’s theory for RH.At this point, Schrödinger knew wave equation approach was working.After determining full wave function for H-like atom, next step is solutions formulti-electron atoms where Bohr’s theory had failed.
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
The Radial Wave Function, R(r)
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Normalizing the Radial Wave FunctionSolutions, Lk
j (𝜌), are called associated Laguerre polynomials.From these, we construct radial part of wave function.Need to retrace steps from u(𝜌) back to R(r) starting with
un,𝓁(𝜌n) = 𝜌𝓁+1n L2𝓁+1
n−𝓁−1(𝜌n) e−𝜌n∕2
going back through
Rn,𝓁(𝜌n) =un,𝓁(𝜌n)𝜌n
= 𝜌𝓁n L2𝓁+1n−𝓁−1(𝜌n) e−𝜌n∕2
and finally with 𝜌 = 2𝜅r returning to
Rn,𝓁(r) = (2𝜅nr)𝓁 L2𝓁+1n−𝓁−1(2𝜅nr) e−𝜅nr
Radial part needs to be normalized, so we redefine Rn,𝓁(r) as
Rn,𝓁(r) = An,𝓁 (2𝜅nr)𝓁 L2𝓁+1n−𝓁−1(2𝜅nr) e−𝜅nr
where An,𝓁 is to-be-determined normalization factor.P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Normalizing the Radial Wave FunctionTo normalization radial part, we define x = 2𝜅nr, and normalization integral is
∫∞
0R∗
n,𝓁(r)Rn,𝓁(r)r2dr = 1(2𝜅n)3 ∫
∞
0R∗
n,𝓁(x)Rn,𝓁(x)x2dx = 1
In terms of x, radial part isRn,𝓁(x) = An,𝓁x𝓁 L2𝓁+1
n−𝓁−1(x) e−x∕2
Substituting Rn,𝓁(x) into integral gives
A2n,𝓁
(2𝜅n)3 ∫∞
0x2𝓁+2 [L2𝓁+1
n−𝓁−1(x)]2 e−xdx = 1
Look up general integral for associated Laguerre polynomials
∫∞
0xk+1 [Lk
j (x)]2 e−xdx = (2j + k + 1)
(j + k)!j!
we findA2
n,𝓁
(2𝜅n)32n(n + 𝓁)!(n − 𝓁 − 1)!
= 1 or An,𝓁 = (2𝜅n)3∕2
√(n − 𝓁 − 1)!2n(n + 𝓁)!
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Radial Wave FunctionTo further simplify Rn,𝓁(r), we introduce quantity analogous to Bohr radius,
a𝜇 ≡ 4𝜋𝜖0ℏ2
q2e𝜇
and 𝜅n = Zna𝜇
so e−𝜅nr becomes e−Zr∕(na𝜇)
Finally, we obtain radial part of wave function of hydrogen-like atom
Rn,𝓁(r) =(
2Zna𝜇
)𝓁+3∕2√
(n − 𝓁 − 1)!2n(n + 𝓁)!
L2𝓁+1n−𝓁−1
(2Zna𝜇
r)
r𝓁 e−Zr∕(na𝜇)
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Radial part of H-like wave functions, Rn,𝓁(r), for n = 1 to n = 4.
R1,0 = 2(
Za𝜇
)3∕2
e−Zr∕a𝜇
R2,0 = 12√
2
(Za𝜇
)3∕2 (2 − Zr
a𝜇
)e−Zr∕(2a𝜇 )
R2,1 = 12√
6
(Za𝜇
)5∕2
re−Zr∕(2a𝜇 )
R3,0 =√
281
(Za𝜇
)3∕2(
6 − 6(
2Zr3a𝜇
)+(
2Zr3a𝜇
)2)
e−Zr∕(3a𝜇 )
R3,1 = 127
√23
(Za𝜇
)5∕2 (4 −
(2Zr3a𝜇
))re−Zr∕(3a𝜇 )
R3,2 = 281
√215
(Za𝜇
)7∕2
r2 e−Zr∕(3a𝜇 )
R4,0 = 116
(Za𝜇
)3∕2 16
(24 − 36
(Zr
2a𝜇
)+ 12
(Zr
2a𝜇
)2
−(
Zr2a𝜇
)3)
e−Zr∕(4a𝜇 )
R4,1 = 132
√1
15
(Za𝜇
)5∕2 12
(20 − 10
(Zr
2a𝜇
)+(
Zr2a𝜇
)2)
r e−Zr∕(4a𝜇 )
R4,2 = 1384
√1
35
(Za𝜇
)7∕2 (6 −
(Zr
2a𝜇
))r2 e−Zr∕(4a𝜇 )
R4,3 = 1768
√15
(Za𝜇
)9∕2
r3 e−Zr∕(4a𝜇 )
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Hydrogen-Like Atom Wave Functions
In summary, solutions to Schrödinger equation for single electron bound to positive chargenucleus are
𝜓n,𝓁,m𝓁(r, 𝜃, 𝜙) = Rn,𝓁(r)Y𝓁,m𝓁
(𝜃, 𝜙).
Rn,𝓁(r) is radial part of wave function and depends on quantum numbers n and 𝓁
Y𝓁,m𝓁(𝜃, 𝜙) is angular part of wave function and depends on quantum numbers 𝓁 and m𝓁.
𝜓n,𝓁,m𝓁(r, 𝜃, 𝜙) called an orbital since it describes electron’s “orbit” around nucleus.
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Hydrogen wave functions, 𝜓n,𝓁,m𝓁(r, 𝜃, 𝜙), for n = 1 to n = 3
Orbital Wave Function
1s 𝜓1,0,0 = 2√4𝜋
(Za𝜇
)3∕2
e−Zr∕a𝜇
2s 𝜓2,0,0 = 1√32𝜋
(Za𝜇
)3∕2 (2 − Zr
a𝜇
)e−Zr∕(2a𝜇 )
2p0 𝜓2,1,0 = 1√32𝜋
(Za𝜇
)5∕2
re−Zr∕(2a𝜇 ) cos 𝜃
2p±1 𝜓2,1,±1 = ∓ 1√64𝜋
(Za𝜇
)5∕2
re−Zr∕(2a𝜇 ) sin 𝜃 e±i𝜙
Orbital Wave Function
3s 𝜓3,0,0 = 1√162𝜋
(Za𝜇
)3∕2(
6 − 6(
2Zr3a𝜇
)+(
2Zr3a𝜇
)2)
e−Zr∕(3a𝜇 )
3p0 𝜓3,1,0 = 127
1√2𝜋
(Za𝜇
)5∕2 (4 −
(2Zr3a𝜇
))re−Zr∕(3a𝜇 ) cos 𝜃
3p±1 𝜓3,1,±1 = ∓ 127
1√4𝜋
(Za𝜇
)5∕2 (4 −
(2Zr3a𝜇
))re−Zr∕(3a𝜇 ) sin 𝜃 e±i𝜙
3d0 𝜓3,2,0 = 281
1√6𝜋
(Za𝜇
)7∕2
r2 e−Zr∕(3a𝜇 ) 12(3 cos2 𝜃 − 1)
3d±1 𝜓3,2,±1 = ∓ 1243
1√𝜋
(Za𝜇
)7∕2
r2 e−Zr∕(3a𝜇 ) 3 cos 𝜃 sin 𝜃 e±i𝜙
3d±2 𝜓3,2,±2 = 1486
1√𝜋
(Za𝜇
)7∕2
r2 e−Zr∕(3a𝜇 ) 3 sin2 𝜃 e±i2𝜙
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Hydrogen-Like Atom Probability distributions
Probability density for finding e− in given volume element, d𝜏, is
|𝜓(r, 𝜃, 𝜙)|2d𝜏 = |Rn,𝓁(r)|2r2dr |Y𝓁,m𝓁(𝜃, 𝜙)|2 sin 𝜃d𝜃d𝜙
Integrate over all values of 𝜃 and 𝜙 to get probability of finding e− inside spherical shell ofthickness dr at distance r from origin
|Rn,𝓁(r)|2r2dr ∫𝜋
0sin 𝜃d𝜃 ∫
2𝜋
0d𝜙 |Y𝓁,m𝓁
(𝜃, 𝜙)|2 = r2R2n,𝓁(r)dr
Recall that spherical harmonic functions are already normalized.
Probability density r2R2n,𝓁(r)dr is called radial distribution function.
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Hydrogen-Like Atom Wave Functions
2D cross sections through x-z planeof H-atom 3D probability densities,labeled as (n,𝓁,m𝓁)Probability goes to zero at wavefunction nodes:
▶ # radial nodes is n − 𝓁 − 1▶ # angular nodes is 𝓁
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Rn,𝓁(r) for s orbitals (𝓁 = 0) of n = 1, 2, 3, 4, 5
20 4 6 8 10 12 14
0.5
0.0
1.0
1.5
2.01s
50 10 15 20 25 30
0.2
0.0
0.4
0.6 2s
10 20 30 40
0.1
0.0
0.2
0.3
0.43s
100 20 30 40 50 60
0.050.00
0.100.150.200.25
4s
100 20 30 40 50 60 70
0.05
0.00
0.10
0.15 5s
20 4 6 8 10 12 14
0.10.0
0.20.30.40.5 1s
0 5 10 15 20 25 30
0.05
0.00
0.10
0.15
0.202s
100 20 30 40 50 60
0.010.00
0.020.030.040.050.06 4s
100 20 30 40
0.020.040.060.080.10 3s
100 20 30 40 50 60 70
0.01
0.00
0.02
0.03
0.04 5s
Wave function extends further out in r, away fromthe nucleus, as n increasesAs with harmonic oscillator classically excludedpositions are displacements where E > V(r)For hydrogen atom, classically excluded radii are
r∕a𝜇 > 2n2
Classically excluded regions are indicated by grayregions.
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Rn,𝓁(r) for n = 5 and all possible values of 𝓁
0.00
0.01
0.02
0.03
0.04
0.05
0.00
0.01
0.02
0.03
0.04
0.05
0.00
0.01
0.02
0.03
0.04
0.05
0.00
0.01
0.02
0.03
0.04
0.05
0 20 40 60 800.00
0.01
0.02
0.03
0.04
0.05
0 20 40 60 80
-0.05
0.00
0.05
0.10
0.15
-0.04
-0.02
0.00
0.02
0.04
-0.04
-0.02
0.00
0.02
0.04
-0.04
-0.02
0.00
0.02
0.04
-0.04
-0.02
0.00
0.02
0.04
5s
5p
5d
5f
5g
5s
5p
5d
5f
5g
Only for s states, 𝓁 = 0, is radial function non-zeroat origin, r = 0.Wave function maximum at constant n is pushedfurther away from nucleus as 𝓁 increases.This is consequence of centrifugal term in effectivepotential.
V̂eff(r) =ℏ2
2𝜇𝓁(𝓁 + 1)
r2⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟
Centrifugal Term
−Zq2
e4𝜋𝜖0r
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Cartesian (real) wave functionsm𝓁 ≠ 0 complex wave function are hard to visualize.Obtain real wave functions by taking sum and difference of degenerate |m𝓁| wavefunctions,
𝜓 (±) = c(𝜓|m𝓁| ± 𝜓−|m𝓁|
)Since 𝜓∗|m𝓁| = 𝜓−|m𝓁| coefficient c is adjusted to make 𝜓 (±) real and normalized.
𝜓px= 1√
2
(𝜓p+1
+ 𝜓p−1
)and 𝜓py
= 1√2i
(𝜓p+1
− 𝜓p−1
)Superposition principle tells us that these are also solutions to same wave equation.If stationary states in linear combination are degenerate, then linear combination isstationary state.Linear combinations of atomic orbitals are important in understanding covalent bonding.Illustrations of Cartesian H-orbital shapes are found in every elementary chemistry text.
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Cartesian (real) wave functions: n = 2, 𝓁 = 1
2px, 2py, 2pzOrbitron Web Site
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Cartesian (real) wave functions
Can also form real wave functions with degenerate d orbitals
ndz2 = 𝜓n,2,0
ndxz =1√2
(𝜓n,2,1 + 𝜓n,2,−1
)ndyz =
1√2i
(𝜓n,2,1 − 𝜓n,2,−1
)ndx2−y2 = 1√
2
(𝜓n,2,2 + 𝜓n,2,−2
)ndxy =
1√2i
(𝜓n,2,2 − 𝜓n,2,−2
)
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Cartesian (real) wave functions: n = 3, 𝓁 = 2
Top: 2dx2−y2 and 2dz2
Bottom 2dxy, 2dxz, and 2dyz
Orbitron Web Site
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom
Web Apps by Paul Falstad
Hydrogen atom orbitals
P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom