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Chapter 18 Electronic Structure of the Hydrogen Atom

Chapter 18 Electronic Structure of the

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 solved analytically and compared to experimental measurements.

Analytical solution serve as basis for obtaining approximate solutions for multi- and , where no analytical solution exists.

Warning: Working through analytical solution of H-atom may cause drowsiness. Do not operate heavy machinery during this derivation.

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Two Particle Problem ⃗ ⃗ 2 particles with mass mp and me at rp and re. z Total mass, center of mass, and inter-particle distance vector is ⃗ ⃗ mprp + mere M = m + m , R⃗ = , and ⃗r = ⃗r − ⃗r p e M e p

y ⃗ ⃗ ⃗ ⃗ Express rp and re in terms of M, R and r as

m mp ⃗r = R⃗ − e ⃗r and ⃗r = R⃗ + ⃗r x p M e M Express individual momenta of 2 particles as ⃗ drp d⃗r p⃗ = m and p⃗ = m e p p dt e e dt then in terms of M, R⃗ and ⃗r as ( ) ( ) ⃗ ⃗ dR m d⃗r dR mp d⃗r p⃗ = m − e and p⃗ = m + p p dt M dt e e dt M dt P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Two Particle Problem Next, consider total energy ⃗ 2 2 pp p⃗ E = + e + V(⃗r) 2mp 2me V(⃗r) depends only on distance between 2 particles. Write energy in terms of M, R⃗ and ⃗r, and obtain ( ) ( ) 2 2 1 dR⃗ 1 d⃗r E = M + 휇 + V(⃗r) 2 dt 2 dt 휇 is reduced mass, given by 1 1 1 휇 = + mp me Define 2 new momenta associated with center of mass and reduced mass, dR⃗ d⃗r p⃗ = M , and p⃗ = 휇 R dt r dt

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Two Particle Problem

Total energy becomes p⃗ 2 p⃗ 2 E = R + r + V(⃗r) 2M 2휇

1st term is translational energy of center of mass

2nd term is kinetic energy due to relative motion of 2 particles

Translating to , we write time independent Schrödinger equation for 2 particle system as [ ] [ ] p⃗ 2 p⃗ 2 ℏ2 ℏ2 R + r + V(⃗r) f (R⃗, ⃗r) = − ∇⃗ 2 − ∇⃗ 2 + V(⃗r) f (R⃗, ⃗r) = Ef (R⃗, ⃗r) 2M 2휇 2M R 2휇 r

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Two Particle Problem [ ] ℏ2 ℏ2 − ∇⃗ 2 − ∇⃗ 2 + V(⃗r) f (R⃗, ⃗r) = Ef (R⃗, ⃗r) 2M R 2휇 r can be separated into product of two wave functions f (R⃗, ⃗r) = 휒(R⃗)휓(⃗r) 휒(R⃗) depending only on center of mass 휓(⃗r) depending only on relative motion of 2 particles Substitute product into Schrödinger Eq above, we obtain 2 wave equations [ ] ℏ2 ℏ2 − ∇⃗ 2 휒(R⃗) = E 휒(R⃗) and − ∇⃗ 2 + V(⃗r) 휓(⃗r) = E 휓(⃗r) 2M R R 2휇 r r

where E = ER + Er On left is wave equation for translational motion of free particle of mass M On right is wave equation for particle with mass 휇 in potential V(⃗r) For electron bound positively charged nucleus, we focus on PDE for 휓(⃗r) 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 휓(⃗r): Electron bound to positively charged nucleus [ ] ℏ2 − ∇2 + V̂ (r) 휓(⃗r) = E휓(⃗r) 2휇 Coulomb potential is

2 ̂ Zqe V(r) = − 휋휖 4 0r

Central potential with 1∕r dependence. V → −∞ as r → 0 V → 0 as r → ∞

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Schrödinger Equation in Spherical Coordinates Since V̂ (r) only depends on r, we adopt spherical coordinates [ ( ( ) )] ℏ2 휕2 2 휕 1 1 휕 휕 1 휕2 − + + sin 휃 + 휓 + V̂ (r)휓 = E휓 2휇 휕r2 r 휕r r2 sin 휃 휕휃 휕휃 sin2 휃 휕휙2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ −L̂ 2∕ℏ2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ ∇2

Take term in parentheses as −L̂ 2∕ℏ2, rearrange, and simplify to

휕2휓(r, 휃, 휙) 휕휓(r, 휃, 휙) 2휇r2 ( ) 1 r2 + 2r + E − V̂ (r) 휓(r, 휃, 휙) = L̂ 2휓(r, 휃, 휙) 휕r2 휕r ℏ2 ℏ2 To solve PDE use separation of variables

휓(r, 휃, 휙) = R(r)Y(휃, 휙)

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Schrödinger Equation in Spherical Coordinates Substitute 휓(r, 휃, 휙) into PDE, and dividing both sides by 휓(r, 휃, 휙)

r2 휕2R(r) 2r 휕R(r) 2휇r2 ( ) 1 1 + + E − V̂ (r) = L̂ 2Y(휃, 휙) R(r) 휕r2 R(r) 휕r ℏ2 Y(휃, 휙) ℏ2 Left side depends only on r Right side depends only on 휃 and 휙. We have turned one PDE into two ODEs. ̂ 1 We know eigenfunctions and eigenvalues of L2 on right side,

̂ 2 휃, 휙 퓁 퓁 ℏ2 휃, 휙 L Y퓁,m( ) = ( + 1) Y퓁,m( )

2 Define 퓁(퓁 + 1) as separation constant and obtain ODE for radial part

r2 d2R(r) 2r dR(r) 2휇r2 ( ) + + E − V̂ (r) = 퓁(퓁 + 1) R(r) dr2 R(r) dr ℏ2

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Schrödinger Equation in Spherical Coordinates Expanding and rearranging ODE for radial part, we obtain [ ( ) ] ℏ2 d2 2 d ℏ2 퓁(퓁 + 1) − + + + V̂ (r) R(r) = ER(r) 2휇 휕r2 r dr 2휇 r2 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

Veff(r)

Looks like 1D time independent Schrödinger Eq with effective potential

ℏ2 퓁(퓁 + 1) Zq2 V̂ (r) = − e eff 2휇 r2 4휋휖 r ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ 0 Centrifugal Term √ , 퓁(퓁 + 1)ℏ, creates centrifugal force that pushes electron away from nucleus. ̂ Notice 2 terms in Veff(r) always have opposite signs. P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom 퓁 Plot of Veff for = 0, 1, 2, and 3.

0.2

0.1

0.0

-0.1

-0.2

-0.3

-0.4 0 2 4 6 8 10 12 14

Note: potential minimum shifts to higher radii with increasing angular momentum (i.e., centrifugal force). P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom ̂ Veff(r): Effective Potential of H-Like Atom Switching to Atomic Units: Convenient units for

Atomic unit of length, also called , is defined as

4휋휖 ℏ2 a ≡ 0 = 52.9177210526763 pm 0 2 qeme

Atomic unit of energy is

ℏ2 q2 1E ≡ = e = 27.21138602818051 eV h 2 4휋휖 a mea0 0 0

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Wave Equation for the Radial Part

[ ( ) ] 2 ℏ2 d2 2 d ℏ2 퓁(퓁 + 1) Zq − + + − e R(r) = ER(r) 휇 2 휇 2 휋휖 2 휕r r dr 2 r 4 0r

pour yourself a fifth cup of coffee...

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation To obtain radial wave function, R(r), we must solve [ ( ) ] 2 ℏ2 d2 2 d ℏ2 퓁(퓁 + 1) Zq − + + − e R(r) = ER(r) 휇 2 휇 2 휋휖 2 휕r r dr 2 r 4 0r Begin by dividing and both sides by E, and obtain [ ] ( ) 2 ℏ2 d2 2 d ℏ2 퓁(퓁 + 1) Zq − + R(r) + − e − 1 R(r) = 0 휇 2 휇 2 휋휖 2 E 휕r r dr 2 E r 4 0rE Define 2휇E ℏ2휅2 휅2 ≡ − or E = − ℏ2 2휇 휅 is in wave numbers. Rearrange potential energy expression to 2 2 2 Zq Zq 2휇 Zq 휇 1 2휌 − e = e = e = 0 휋휖 휋휖 ℏ2휅2 휋휖 ℏ2휅 휅 휅 4 0rE 4 0r 2 0 r r Zq2휇 휌 is a dimensionless quantity: 휌 ≡ e 0 0 휋휖 ℏ2휅 2 0 P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation Substitute expression for potential energy into ODE [ ] ( ) 1 d2 2 d 퓁(퓁 + 1) 2휌 + R(r) − 1 + + 0 R(r) = 0 휅2 휕r2 r dr (휅r)2 휅r Define dimensionless variable: 휌 ≡ 2휅r Rewrite radial part as function of 휌 as [ ] ( ) d2 2 d 퓁(퓁 + 1) 2휌 + R(휌) − 1 + + 0 R(휌) = 0 d휌2 휌 d휌 휌2 휌 Further simplify ODE by defining du(휌) dR(휌) d2u(휌) d2R(휌) dR(휌) u(휌) = 휌R(휌), = 휌 + R(휌), = 휌 + 2 d휌 d휌 d휌2 d휌2 d휌

Re-express ODE as ( ) d2u(휌) 1 휌 퓁(퓁 + 1) − − 0 + u(휌) = 0 d휌2 4 2휌 휌2 휌 varies from 0 to ∞. 1st look at asymptotic solutions: (1) 휌 → ∞ and (2) 휌 → 0 P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation, 휌 → ∞

Starting with ( ) d2u(휌) 1 휌 퓁(퓁 + 1) − − 0 + u(휌) = 0 d휌2 4 2휌 휌2 (1) 휌 → ∞. At large values of 휌 approximate ODE as

d2u(휌) u(휌) − ≈ 0 d휌2 4

Solutions to this ODE have form u(휌) ∼ Ae−휌∕2 + Be휌∕2

Reject positive exponent since require that solution be finite everywhere,

u(휌) ∼ Ae−휌∕2 in limit that 휌 → ∞

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation, 휌 → 0

Starting with ( ) d2u(휌) 1 휌 퓁(퓁 + 1) − − 0 + u(휌) = 0. d휌2 4 2휌 휌2 (2) 휌 → 0. At small values of 휌 approximate ODE as

d2u(휌) 퓁(퓁 + 1) − u(휌) ≈ 0. d휌2 휌2

Solutions to this ODE have form u(휌) ∼ A휌퓁+1 + B휌−퓁.

Reject B term again because require that solution be finite at 휌 = 0.

u(휌) ∼ A휌퓁+1 in limit that 휌 → 0

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation

u(휌) ∼ Ae−휌∕2 in limit that 휌 → ∞ u(휌) ∼ A휌퓁+1 in limit that 휌 → 0

Now, back to ( ) d2u(휌) 1 휌 퓁(퓁 + 1) − − 0 + u(휌) = 0 d휌2 4 2휌 휌2

we propose general solution of form

u(휌) = 휌퓁+1 L(휌) e−휌∕2

This has correct behavior in two limits. Only need to determine L(휌) to get behavior for all 휌.

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation Plug u(휌) = 휌퓁+1 L(휌) e−휌∕2, where 휌 ≡ 2휅r, into ( ) d2u(휌) 1 휌 퓁(퓁 + 1) − − 0 + u(휌) = 0 d휌2 4 2휌 휌2

gives ( ) d2L(휌) ( )dL(휌) 휌 휌 + 2퓁 + 2 −휌 + 0 − (퓁 + 1) L(휌) = 0 휌2 ⏟⏟⏟ 휌 d d ⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟2 k+1 j 휌 퓁 퓁 Set j = 0∕2 − ( + 1), and k = 2 + 1 and this ODE is recognized as 2 k 휌 k 휌 d Lj ( ) dLj ( ) 휌 + (k + 1 − 휌) + jLk(휌) = 0, associated Laguerre differential equation d휌2 d휌 j Has nonsingular solutions only if j is non-negative integers, j = 0, 1, 2, … k 휌 These solutions, Lj ( ), are called the associated Laguerre polynomials. P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Selected associated Laguerre polynomials 퓁 2퓁+1 휌 Polynomial Roots n Ln−퓁−1( ) 1 휌 1 0 L0( ) 1 - 1 휌 휌 2 0 L1( ) 2 − 2 - - 2 1 L3(휌) 1 - - 0 ( ) 3 0 L1(휌) 1 6 − 6휌 + 휌2 1.26795 4.73205 - 2 2 3 휌 휌 3 1 L1( ) 4 − 4 - - 3 2 L5(휌) 1 - - - 0 ( ) 4 0 L1(휌) 1 24 − 36휌 + 12휌2 − 휌3 7.75877 0.935822 3.30541 3 6 ( ) 4 1 L3(휌) 1 20 − 10휌 + 휌2 2.76393 7.23607 - 2 2 5 휌 휌 4 2 L1( ) 6 − 6 - - 4 3 L7(휌) 1 - - - 0 ( ) 5 0 L1(휌) 1 120 − 240휌 + 120휌2 − 20휌3 + 휌4 0.743292 2.57164 5.73118 10.9539 4 24 ( ) 5 1 L3(휌) 1 120 − 90휌 + 18휌2 − 휌3 2.14122 5.31552 10.5433 - 3 6 ( ) 5 2 L5(휌) 1 42 − 14휌 + 휌2 4.35425 9.64575 - - 2 2 7 휌 휌 5 3 L1( ) 8 − 8 - - - 9 휌 5 4 L0( ) 1 - - - - dk Associated Laguerre polynomials following definition where Lk(x) = (−1)k L (x). j dxk j+k 1 dj ( ) Laguerre polynomial L (x) is defined by Rodrigues formula: L (x) = ex xje−x . j j n! dxj P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation Since 퓁 = 0, 1, 2, … , and j = 0, 1, 2, … , then 휌 0 = j + 퓁 + 1 2 휌 , , can only take on integer values of 0∕2 = 1 2 …. We define this as the principal ≡ 휌 퓁 , n 0∕2 = j + + 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 value KLMNO ← symbol

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation 퓁 is called the Rearranging n = j + 퓁 + 1 for 퓁 gives

퓁 = n − j − 1

and we find that 퓁 cannot exceed n − 1 (since lowest value of j is zero). Range of 퓁 is 퓁 = 0, … , n − 1 √ Recall that azimuthal quantum number, 퓁, defines total angular momentum of 퓁(퓁 + 1)ℏ.

m퓁 is called the .

m퓁 has positive and negative integer values between −퓁 and 퓁. When x, y, or z component of the electron’s angular momentum is measured, only values of m퓁ℏ are observed.

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Solving the Radial Wave Equation Wave functions with same value of n and 퓁 form set called a sub-shell. Special letters are assigned to sub-shell with given 퓁 value, 퓁 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ← numerical value spdfghiklmoq r t ← symbol and continue afterwards in alphabetical order. 1st four letters arose in pre-quantum atomic spectroscopy for classifying emission lines and stand for sharp, principal, diffuse, and fine. Shorthand n퓁 notation uses with 퓁 symbol, Wave function with ▶ n = 1, 퓁 = 0 is referred to as 1s state ▶ n = 2, 퓁 = 1 is referred to as 2p state Number of roots of L2퓁+1 is n 퓁 . Thus, number of radial nodes is equal to n 퓁 . n−퓁−1 − − 1 − − 1 Recall for that number of angular nodes is 퓁 Thus, total number of nodes is (n − 퓁 − 1) + 퓁 = n − 1.

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Energy of the Hydrogen-Like Atom 휌 Given the constraint that 0∕2 = n we go back to Zq2휇 Zq2휇 휌 ≡ e = 2n and rearrange to 휅 = e 0 휋휖 ℏ2휅 n 휋휖 ℏ2 2 0 4 0 n

From 2휇E ℏ2휅2 Z2q4휇 Z2q4휇 휅2 ≡ − rearranges to E = − n = − e = − e ℏ2 n 2휇 휋2휖2ℏ2 2 휖2 2 2 32 0 n 8 0h n Z2q4휇 E = − e Energy of H-like atom n 휖2 2 2 8 0h n

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Energy of the Hydrogen-Like Atom

Z2q4휇 E = − e Energy of H-like atom n 휖2 2 2 8 0h n

e− bound to nucleus with charge Z energy only depends on n and Z If Z increases (holding n constant) then En decreases—becomes more negative. Higher Z means nucleus holds e− more tightly In limit that n → ∞ then E → 0 and e− is unbound

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Energy of the Hydrogen-Like Atom 휇 With ≈ me, the n = 1 energy is called Rydberg unit of energy (Ry) 4 q me 1Ry = −E = e = 13.60569301218355 eV 1 휖2 2 8 0h 휇 With ≈ me approximation 1 Ry E = − Energy of H atom n n2

Can also divide by atomic unit of energy, Eh, to obtain

1 E E = − h Energy of H atom n 2n2

Given n, energy is identical for each 퓁. Given 퓁 energy is identical for 2퓁 + 1 values of m퓁.

∑n−1 ∑n−1 퓁 퓁 2 Degeneracy of nth is gn = (2 + 1) = n + 2 = n + n(n − 1) = n 퓁=0 퓁=0

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Energy levels of single electron bound to

orbitals degeneracy

Visible light emission ”

UV light emission “Lyman series

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Energy of the Hydrogen-Like Atom Remember Balmer series for hydrogen emission spectra obeys relation: ( ) 1 1 1 = R − where R = 1.097 × 107 /m 휆 H 22 n2 H Followed by Bohr’s theory of atom which gave ( ) ( ) ( ) 2 4 1 1 meq 1 1 = e Z2 − 휆 휋휖 휋ℏ3 2 2 4 0 4 c0 n n ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ f i

perfect agreement with RH

휈 h휈 휆 Taking h = 휆 = ΔE = Ef − Ei and solving for 1∕ 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 for multi-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 Function k 휌 Solutions, Lj ( ), are called associated Laguerre polynomials. From these, we construct radial part of wave function. Need to retrace steps from u(휌) back to R(r) starting with 퓁 퓁 휌 휌 휌 +1 2 +1 휌 − n∕2 un,퓁( n) = n Ln−퓁−1( n) e going back through 휌 un,퓁( n) 퓁 퓁 휌 R (휌 ) = = 휌 L2 +1 (휌 ) e− n∕2 n,퓁 n 휌 n n−퓁−1 n n and finally with 휌 = 2휅r returning to 퓁 퓁 휅 휅 2 +1 휅 − nr Rn,퓁(r) = (2 nr) Ln−퓁−1(2 nr) e

Radial part needs to be normalized, so we redefine Rn,퓁(r) as 퓁 퓁 휅 휅 2 +1 휅 − nr Rn,퓁(r) = An,퓁 (2 nr) Ln−퓁−1(2 nr) e

where An,퓁 is to-be-determined normalization factor. P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Normalizing the Radial Wave Function 휅 To normalization radial part, we define x = 2 nr, and normalization integral is ∞ ∞ ∗ 2 1 ∗ 2 R (r)R ,퓁(r)r dr = R (x)R ,퓁(x)x dx = 1 ∫ n,퓁 n 휅 3 ∫ n,퓁 n 0 (2 n) 0 In terms of x, radial part is R x A x퓁 L2퓁+1 x e−x∕2 n,퓁( ) = n,퓁 n−퓁−1( ) Substituting Rn,퓁(x) into integral gives A2 ∞ n,퓁 x2퓁+2 [L2퓁+1 (x)]2 e−xdx = 1 휅 3 ∫ n−퓁−1 (2 n) 0 Look up general integral for associated Laguerre polynomials ∞ k+1 k 2 −x (j + k)! ∫ x [Lj (x)] e dx = (2j + k + 1) 0 j! √ A2 퓁 퓁 n,퓁 2n(n + )! 3∕2 (n − − 1)! we find = 1 or A ,퓁 = (2휅 ) 휅 3 퓁 n n 퓁 (2 n) (n − − 1)! 2n(n + )!

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Radial Wave Function

To further simplify Rn,퓁(r), we introduce quantity analogous to Bohr radius, 휋휖 ℏ2 4 0 Z a휇 ≡ and 휅 = 2휇 n qe na휇 휅 so e− nr becomes e−Zr∕(na휇) Finally, we obtain radial part of wave function of hydrogen-like atom

( ) √ ( ) 퓁+3∕2 퓁 2Z (n − − 1)! 2퓁+1 2Z 퓁 −Zr∕(na휇) Rn,퓁(r) = Ln−퓁−1 r r e na휇 2n(n + 퓁)! na휇

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Radial part of H-like wave functions, Rn,퓁(r), for n = 1 to n = 4.

( )3∕2 Z −Zr∕a휇 R1,0 = 2 e a휇 ( ) ( )3∕2 ( ) ( )3∕2 ( ) ( )2 ( )3 1 Z Zr −Zr∕(2a휇 ) 1 Z 1 Zr Zr Zr −Zr∕(4a휇 ) R2,0 = √ 2 − e R4,0 = 24 − 36 + 12 − e 2 2 a휇 a휇 16 a휇 6 2a휇 2a휇 2a휇 √ ( ) ( )5∕2 ( )5∕2 ( ) ( )2 1 Z −Zr∕(2a휇 ) 1 1 Z 1 Zr Zr −Zr∕(4a휇) R2,1 = √ re R4,1 = 20 − 10 + r e 2 6 a휇 32 15 a휇 2 2a휇 2a휇 √ ( ) √ ( ) ( ( )) ( )3∕2 ( ) ( )2 7∕2 2 Z 2Zr 2Zr −Zr∕(3a휇 ) 1 1 Z Zr 2 −Zr∕(4a휇 ) R3,0 = 6 − 6 + e R4,2 = 6 − r e 81 a휇 3a휇 3a휇 384 35 a휇 2a휇

√ √ ( ) ( )5∕2 ( ( )) 9∕2 1 2 Z 2Zr −Zr∕(3a휇 ) 1 1 Z 3 −Zr∕(4a휇 ) R3,1 = 4 − re R4,3 = r e 27 3 a휇 3a휇 768 5 a휇 √ ( )7∕2 2 2 Z 2 −Zr∕(3a휇 ) R3,2 = r e 81 15 a휇

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 charge nucleus are 휓 , 휃, 휙 휃, 휙 . n,퓁,m퓁 (r ) = Rn,퓁(r)Y퓁,m퓁 ( )

퓁 Rn,퓁(r) is radial part of wave function and depends on quantum numbers n and

휃, 휙 퓁 Y퓁,m퓁 ( ) is angular part of wave function and depends on quantum numbers and m퓁.

휓 , 휃, 휙 n,퓁,m퓁 (r ) called an orbital since it describes electron’s “orbit” around nucleus.

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom 휓 , 휃, 휙 Hydrogen wave functions, n,퓁,m퓁 (r ), for n = 1 to n = 3

Orbital Wave Function ( ) ( )3∕2 ( ) ( )2 휓 1 Z 2Zr 2Zr −Zr∕(3a휇) 3s 3,0,0 = √ 6 − 6 + e Orbital Wave Function 162휋 a휇 3a휇 3a휇 ( )3∕2 ( )5∕2 ( ( )) 휓 √2 Z −Zr∕a휇 1 1 Z 2Zr 1s 1,0,0 = e 3p 휓 = √ 4 − re−Zr∕(3a휇) cos 휃 휋 a휇 0 3,1,0 4 27 2휋 a휇 3a휇 ( ) ( ( )) ( )3∕2 ( ) 5∕2 휓 1 Z Zr −Zr∕(2a휇) 휓 1 1 Z 2Zr −Zr∕(3a휇 ) 휃 ±i휙 2s 2,0,0 = √ 2 − e 3p±1 3,1,±1 = ∓ √ 4 − re sin e 32휋 a휇 a휇 27 4휋 a휇 3a휇 ( ) ( )5∕2 7∕2 휓 1 Z −Zr∕(2a휇 ) 휃 휓 2 1 Z 2 −Zr∕(3a휇) 1 2 휃 2p0 2,1,0 = √ re cos 3d0 3,2,0 = √ r e (3 cos − 1) 32휋 a휇 81 6휋 a휇 2 ( ) ( )5∕2 7∕2 휓 1 Z −Zr∕(2a휇 ) 휃 ±i휙 휓 1 1 Z 2 −Zr∕(3a휇 ) 휃 휃 ±i휙 2p±1 2,1,±1 = ∓ √ re sin e 3d±1 3,2,±1 = ∓ √ r e 3 cos sin e 64휋 a휇 243 휋 a휇

( )7∕2 휓 1 1 Z 2 −Zr∕(3a휇) 2 휃 ±i2휙 3d±2 3,2,±2 = √ r e 3 sin e 486 휋 a휇

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, d휏, is

|휓 , 휃, 휙 |2 휏 | |2 2 | 휃, 휙 |2 휃 휃 휙 (r ) d = Rn,퓁(r) r dr Y퓁,m퓁 ( ) sin d d

Integrate over all values of 휃 and 휙 to get probability of finding e− inside spherical shell of thickness dr at distance r from origin

휋 2휋 |R (r)|2r2dr sin 휃d휃 d휙 |Y (휃, 휙)|2 = r2R2 (r)dr n,퓁 ∫ ∫ 퓁,m퓁 n,퓁 0 0 Recall that spherical harmonic functions are already normalized.

Probability density 2 2 is called radial distribution function. r Rn,퓁(r)dr

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Hydrogen-Like Atom Wave Functions

2D cross sections through x-z plane of H-atom 3D probability densities, labeled as (n, 퓁, m퓁) Probability goes to zero at wave function nodes: ▶ # radial nodes is n − 퓁 − 1 ▶ # angular nodes is 퓁

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom 퓁 , , , , Rn,퓁(r) for s orbitals ( = 0) of n = 1 2 3 4 5 2.0 1s 0.5 1s 1.5 0.4 1.0 0.3 0.2 0.5 0.1 0.0 0.0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0.20 0.6 2s 2s 0.15 Wave function extends further out in r, away from 0.4 0.10 0.2 the nucleus, as n increases 0.05 0.0 0.00 0 5 10 15 20 25 30 0 5 10 15 20 25 30 As with harmonic oscillator classically excluded 0.4 0.10 3s 3s 0.3 0.08 positions are displacements where E > V(r) 0.2 0.06 0.1 0.04 0.0 0.02 For hydrogen atom, classically excluded radii are

10 20 30 40 0 10 20 30 40 0.25 4s 0.06 4s 0.20 0.05 > 2 0.15 0.04 r∕a휇 2n 0.10 0.03 0.05 0.02 0.00 0.01 0.00 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Classically excluded regions are indicated by gray 0.15 5s 0.04 5s 0.10 0.03 regions. 0.05 0.02 0.01 0.00 0.00 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom 퓁 Rn,퓁(r) for n = 5 and all possible values of 0.05 0.15 5s 0.04 5s 0.10 0.03 0.05 0.02 0.00 0.01 -0.05 0.00 0.05 퓁 0.04 Only for s states, = 0, is radial function non-zero 5p 0.04 5p 0.02 0.03 0.00 at origin, r = 0. 0.02 -0.02 0.01 -0.04 0.00 Wave function maximum at constant n is pushed 0.05 0.04 5d 0.04 5d further away from nucleus as 퓁 increases. 0.02 0.03 0.00 0.02 This is consequence of centrifugal term in effective -0.02 0.01 -0.04 0.00 potential. 0.05 0.04 5f 0.04 5f 0.02 0.03 2 0.00 2 퓁 퓁 Zq 0.02 ℏ ( + 1) -0.02 ̂ e 0.01 Veff(r) = − -0.04 휇 2 휋휖 0.00 2 r 4 0r 0.05 0.04 ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ 5g 0.04 5g 0.02 0.03 0.00 Centrifugal Term 0.02 -0.02 0.01 -0.04 0.00 0 20 40 60 80 0 20 40 60 80

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Cartesian (real) wave functions

m퓁 ≠ 0 complex wave function are hard to visualize.

Obtain real wave functions by taking sum and difference of degenerate |m퓁| wave functions, ( ) 휓(±) 휓 휓 = c |m퓁| ± −|m퓁|

∗ (±) Since 휓 = 휓 | | coefficient c is adjusted to make 휓 real and normalized. |m퓁| − m퓁 ( ) ( ) 1 1 휓 = √ 휓 + 휓 and 휓 = √ 휓 − 휓 px p+1 p−1 py p+1 p−1 2 2i

Superposition principle tells us that these are also solutions to same wave equation. If stationary states in linear combination are degenerate, then linear combination is . 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, 퓁 = 1

2px, 2py, 2pz Orbitron 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

휓 ndz2 = n,2,0 ( ) ( ) 1 휓 휓 1 휓 휓 ndxz = √ n,2,1 + n,2,−1 ndyz = √ n,2,1 − n,2,−1 2 2i ( ) ( ) 1 휓 휓 1 휓 휓 ndx2−y2 = √ n,2,2 + n,2,−2 ndxy = √ n,2,2 − n,2,−2 2 2i

P. J. Grandinetti Chapter 18: Electronic Structure of the Hydrogen Atom Cartesian (real) wave functions: n = 3, 퓁 = 2

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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

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