Physical Chemistry · Part 5 of 9 · Free
The Hydrogen Atom — formula sheet
Every key expression and definition from Quantum Chemistry, Part 5, on one page. Free to read, no sign-in.
Key expressions
the separated two-body Hamiltonian
Ĥ = Ĥ_CM + Ĥ_rel = −(ℏ²/2M_tot)∇²_CM − (ℏ²/2μ)∇²_r + V(r)
Ĥ = Ĥ_CM + Ĥ_rel = −(ℏ²/2M_tot)∇²_CM − (ℏ²/2μ)∇²_r + V(r)
reduced mass of an electron and a nucleus of mass M
μ = m_eM / (m_e + M) = m_e / (1 + m_e/M)
μ = m_eM / (m_e + M) = m_e / (1 + m_e/M)
Ĥ = −(ℏ²/2m_1)∇²_1 − (ℏ²/2m_2)∇²_2 + V(|r_2 − r_1|)
R = (m_1r_1 + m_2r_2) / (m_1 + m_2) , r = r_2 − r_1
the kinetic-energy identity behind the reduced mass
(1/m_1)∇²_1 + (1/m_2)∇²_2 = (1/M_tot)∇²_R + (1/μ)∇²_r
(1/m_1)∇²_1 + (1/m_2)∇²_2 = (1/M_tot)∇²_R + (1/μ)∇²_r
the Rydberg constant for a nucleus of finite mass M
R_M = R_∞ · μ/m_e = R_∞ / (1 + m_e/M)
R_M = R_∞ · μ/m_e = R_∞ / (1 + m_e/M)
the Coulomb potential energy of an electron and a nucleus of charge +Ze
V(r) = −Ze² / (4πε_0r)
V(r) = −Ze² / (4πε_0r)
the Schrödinger equation for a central field
−(ℏ²/2μ)∇²ψ + V(r)ψ = Eψ , with ∇² = (1/r²)∂/∂r(r²∂/∂r) − L̂²/(ℏ²r²)
−(ℏ²/2μ)∇²ψ + V(r)ψ = Eψ , with ∇² = (1/r²)∂/∂r(r²∂/∂r) − L̂²/(ℏ²r²)
the spherical harmonics, imported unchanged from Part 4
L̂² Y_l^m(θ,φ) = l(l+1)ℏ² Y_l^m(θ,φ) , L̂_z Y_l^m = m_lℏ Y_l^m
L̂² Y_l^m(θ,φ) = l(l+1)ℏ² Y_l^m(θ,φ) , L̂_z Y_l^m = m_lℏ Y_l^m
(1/r²)∂/∂r(r² ∂ψ/∂r) + (1/r²sinθ)∂/∂θ(sinθ ∂ψ/∂θ) + (1/r²sin²θ)∂²ψ/∂φ² + (2μ/ℏ²)[E − V(r)]ψ = 0
(1/R) d/dr(r² dR/dr) + (2μr²/ℏ²)[E − V(r)] = −(1/Y)Λ̂Y ≡ β
the radial equation for a central field
(1/r²) d/dr(r² dR/dr) + [ (2μ/ℏ²)(E − V(r)) − l(l+1)/r² ] R = 0
(1/r²) d/dr(r² dR/dr) + [ (2μ/ℏ²)(E − V(r)) − l(l+1)/r² ] R = 0
the radial equation in one-dimensional form
−(ℏ²/2μ) d²u/dr² + [ V(r) + ℏ²l(l+1)/(2μr²) ] u = E u
−(ℏ²/2μ) d²u/dr² + [ V(r) + ℏ²l(l+1)/(2μr²) ] u = E u
the hydrogen-atom energy levels
E_n = − (μe⁴ / 8ε_0²h²) · Z²/n² = −13.6057 Z²/n² eV , n = 1, 2, 3, …
E_n = − (μe⁴ / 8ε_0²h²) · Z²/n² = −13.6057 Z²/n² eV , n = 1, 2, 3, …
the origin of the l ≤ n−1 rule
n = n_r + l + 1 , n_r = 0, 1, 2, … ⇒ l ≤ n − 1
n = n_r + l + 1 , n_r = 0, 1, 2, … ⇒ l ≤ n − 1
κ = √(−2μE) / ℏ (real and positive for E < 0), ρ = 2κr , λ = μZe² / (4πε_0ℏ²κ) = Z/(κa_0)
the radial equation in dimensionless form
d²u/dρ² = [ ¼ − λ/ρ + l(l+1)/ρ² ] u
d²u/dρ² = [ ¼ − λ/ρ + l(l+1)/ρ² ] u
u(ρ) = ρ^l+1 e^−ρ/2 v(ρ)
the associated Laguerre equation, α = 2l+1
ρ v″ + (2l + 2 − ρ) v′ + (λ − l − 1) v = 0
ρ v″ + (2l + 2 − ρ) v′ + (λ − l − 1) v = 0
the radial recursion relation
c_j+1 / c_j = (j + l + 1 − λ) / [ (j+1)(j + 2l + 2) ]
c_j+1 / c_j = (j + l + 1 − λ) / [ (j+1)(j + 2l + 2) ]
c_j+1/c_j → j/(j·j) = 1/j as j → ∞
u = ρ^l+1 e^−ρ/2 v ~ ρ^l+1 e^−ρ/2 e^+ρ = ρ^l+1 e^+ρ/2 → ∞
the quantisation condition
λ = n_r + l + 1 ≡ n , n_r = 0, 1, 2, … ⇒ n = 1, 2, 3, … and l = 0, 1, …, n−1
λ = n_r + l + 1 ≡ n , n_r = 0, 1, 2, … ⇒ n = 1, 2, 3, … and l = 0, 1, …, n−1
the hydrogenic energy levels, all three standard forms
E_n = − ℏ²Z² / (2μa_0²n²) = − μZ²e⁴ / (8ε_0²h²n²) = − (Z²/n²) × 13.6057 eV
E_n = − ℏ²Z² / (2μa_0²n²) = − μZ²e⁴ / (8ε_0²h²n²) = − (Z²/n²) × 13.6057 eV
the Bohr radius
a_0 = 4πε_0ℏ² / (μe²) = 52.9177 pm (with m_e in place of μ)
a_0 = 4πε_0ℏ² / (μe²) = 52.9177 pm (with m_e in place of μ)
the hydrogenic wavefunction in closed form
ψ_nlm(r,θ,φ) = N_nl e^−ρ/2 ρ^l L_n−l−1^2l+1(ρ) Y_l^m(θ,φ) , ρ = 2Zr/(na_0)
ψ_nlm(r,θ,φ) = N_nl e^−ρ/2 ρ^l L_n−l−1^2l+1(ρ) Y_l^m(θ,φ) , ρ = 2Zr/(na_0)
the radial normalisation constant (convention-dependent — see the note)
N_nl = [ (2Z/na_0)³ (n−l−1)! / (2n · (n+l)!) ]^½
N_nl = [ (2Z/na_0)³ (n−l−1)! / (2n · (n+l)!) ]^½
the degeneracy of the n-th hydrogenic level
g_n = Σ_l=0^n−1 (2l+1) = 2·[n(n−1)/2] + n = n²
g_n = Σ_l=0^n−1 (2l+1) = 2·[n(n−1)/2] + n = n²
the Laplace–Runge–Lenz vector
M = (1/2μ)(p × L − L × p) − (Ze²/4πε_0) r/r , [Ĥ, M̂] = 0 for V ∝ 1/r
M = (1/2μ)(p × L − L × p) − (Ze²/4πε_0) r/r , [Ĥ, M̂] = 0 for V ∝ 1/r
the ordering within a shell in many-electron atoms
E(ns) < E(np) < E(nd) < E(nf) for every atom except one-electron ones
E(ns) < E(np) < E(nd) < E(nf) for every atom except one-electron ones
the structure of every hydrogenic orbital
ψ_nlm(r,θ,φ) = R_nl(r) · Y_l^m(θ,φ)
ψ_nlm(r,θ,φ) = R_nl(r) · Y_l^m(θ,φ)
the angular momentum of a hydrogenic orbital
|L| = √[l(l+1)] ℏ , L_z = m_lℏ
|L| = √[l(l+1)] ℏ , L_z = m_lℏ
the standard degeneracy counts
orbitals with quantum number l: 2l + 1 orbitals with quantum number n: n² electrons in shell n: 2n²
orbitals with quantum number l: 2l + 1 orbitals with quantum number n: n² electrons in shell n: 2n²
the commuting set that labels hydrogenic states
[Ĥ, L̂²] = 0 , [Ĥ, L̂_z] = 0 , [L̂², L̂_z] = 0
[Ĥ, L̂²] = 0 , [Ĥ, L̂_z] = 0 , [L̂², L̂_z] = 0
the electron spin quantum numbers
s = ½ always ; m_s = +½ or −½ ; |S| = √[s(s+1)]ℏ = (√3/2)ℏ
s = ½ always ; m_s = +½ or −½ ; |S| = √[s(s+1)]ℏ = (√3/2)ℏ
the radial node rule
number of radial nodes = n − l − 1
number of radial nodes = n − l − 1
the hydrogenic Z-scaling rules
a_0 → a_0/Z — so every distance ∝ 1/Z (r_max = n²a_0/Z) , E_n → Z²E_n , R_nl(r) → Z^3/2R_nl(Zr)
a_0 → a_0/Z — so every distance ∝ 1/Z (r_max = n²a_0/Z) , E_n → Z²E_n , R_nl(r) → Z^3/2R_nl(Zr)
the electron density at the nucleus
R_ns(0) = 2(Z/na_0)^3/2 , |ψ_ns(0)|² = (1/π)(Z/na_0)³
R_ns(0) = 2(Z/na_0)^3/2 , |ψ_ns(0)|² = (1/π)(Z/na_0)³
radial orthonormality at fixed l
∫_0^∞ R_nl(r) R_n′l(r) r² dr = δ_nn′
∫_0^∞ R_nl(r) R_n′l(r) r² dr = δ_nn′
the node budget of a hydrogenic orbital
number of angular nodes = l ; total nodal surfaces = (n − l − 1) + l = n − 1
number of angular nodes = l ; total nodal surfaces = (n − l − 1) + l = n − 1
the real p orbitals as combinations of the complex ones
p_x = (1/√2)(ψ_2p,−1 − ψ_2p,+1) , p_y = (i/√2)(ψ_2p,−1 + ψ_2p,+1) , p_z = ψ_2p,0
p_x = (1/√2)(ψ_2p,−1 − ψ_2p,+1) , p_y = (i/√2)(ψ_2p,−1 + ψ_2p,+1) , p_z = ψ_2p,0
the radial distribution function
P(r) = r² [R_nl(r)]² (any orbital) P(r) = 4πr²|ψ|² (s orbitals)
P(r) = r² [R_nl(r)]² (any orbital) P(r) = 4πr²|ψ|² (s orbitals)
the 1s radial distribution function
P(r) = 4r²e^−2r/a₀/a_0³ for the 1s orbital
P(r) = 4r²e^−2r/a₀/a_0³ for the 1s orbital
the volume element in spherical polar coordinates
dτ = r² sinθ dr dθ dφ
dτ = r² sinθ dr dθ dφ
P(r) dr = [ ∫_0^π ∫_0^2π |R_nlY_l^m|² r² sinθ dθ dφ ] dr = r²R_nl² dr ∫|Y|²dΩ = r²R_nl² dr
the expectation value of a radial quantity
⟨A⟩ = ∫ ψ* Â ψ dτ , and for a function of r alone, ⟨f(r)⟩ = ∫_0^∞ f(r) P(r) dr
⟨A⟩ = ∫ ψ* Â ψ dτ , and for a function of r alone, ⟨f(r)⟩ = ∫_0^∞ f(r) P(r) dr
the mean radius of a hydrogenic orbital
⟨r⟩_nl = (a_0/2Z)[3n² − l(l+1)]
⟨r⟩_nl = (a_0/2Z)[3n² − l(l+1)]
the mean inverse radius — depends on n only
⟨1/r⟩_nl = Z/(n²a_0)
⟨1/r⟩_nl = Z/(n²a_0)
the mean square radius
⟨r²⟩_nl = (n²a_0²/2Z²)[5n² + 1 − 3l(l+1)]
⟨r²⟩_nl = (n²a_0²/2Z²)[5n² + 1 − 3l(l+1)]
⟨r⟩ = ∫_0^∞ r · P(r) dr = (4/a_0³) ∫_0^∞ r³ e^−2r/a₀ dr
⟨r⟩ = (4/a_0³) × 6 × (a_0/2)⁴ = (4/a_0³) × 6 × a_0⁴/16 = 3a_0/2
the virial theorem for a Coulombic system
2⟨T⟩ = −⟨V⟩ , E = ⟨T⟩ + ⟨V⟩ = −⟨T⟩ = ½⟨V⟩
2⟨T⟩ = −⟨V⟩ , E = ⟨T⟩ + ⟨V⟩ = −⟨T⟩ = ½⟨V⟩
kinetic and potential energies of a hydrogenic state
⟨T⟩_n = +13.6057 Z²/n² eV , ⟨V⟩_n = −27.2114 Z²/n² eV , E_n = −13.6057 Z²/n² eV
⟨T⟩_n = +13.6057 Z²/n² eV , ⟨V⟩_n = −27.2114 Z²/n² eV , E_n = −13.6057 Z²/n² eV
the photon energy of a hydrogenic transition
hν = E_n₂ − E_n₁ = 13.6057 Z² (1/n₁² − 1/n₂²) eV
hν = E_n₂ − E_n₁ = 13.6057 Z² (1/n₁² − 1/n₂²) eV
the Rydberg formula
ν̃ = 1/λ = R_H Z² (1/n₁² − 1/n₂²) , R_H = 109677.58 cm⁻¹
ν̃ = 1/λ = R_H Z² (1/n₁² − 1/n₂²) , R_H = 109677.58 cm⁻¹
electric-dipole selection rules for a one-electron atom
Δl = ±1 (compulsory) Δm_l = 0, ±1 Δn = anything, including 0
Δl = ±1 (compulsory) Δm_l = 0, ±1 Δn = anything, including 0
the Z-scaling of every hydrogenic quantity
E_n = −13.6057 Z²/n² eV , r ∝ a_0/Z , ν̃ = R Z²(1/n₁² − 1/n₂²)
E_n = −13.6057 Z²/n² eV , r ∝ a_0/Z , ν̃ = R Z²(1/n₁² − 1/n₂²)
the spin–orbit Hamiltonian
Ĥ_SO = ξ(r) L̂·Ŝ , ξ(r) = (1/2m_e²c²)(1/r)(dV/dr)
Ĥ_SO = ξ(r) L̂·Ŝ , ξ(r) = (1/2m_e²c²)(1/r)(dV/dr)
the spin–orbit energy in terms of j
⟨L̂·Ŝ⟩ = (ℏ²/2)[j(j+1) − l(l+1) − s(s+1)]
⟨L̂·Ŝ⟩ = (ℏ²/2)[j(j+1) − l(l+1) − s(s+1)]
the spin–orbit coupling constant and the doublet splitting it produces
ζ_nl ∝ Z⁴/[n³ l(l+½)(l+1)] , ΔE_SO = ζ_nl(l+½) ∝ Z⁴/[n³ l(l+1)]
ζ_nl ∝ Z⁴/[n³ l(l+½)(l+1)] , ΔE_SO = ζ_nl(l+½) ∝ Z⁴/[n³ l(l+1)]
the Dirac fine-structure formula
E_n,j = −(13.6057/n²)[1 + (α²/n²)(n/(j+½) − ¾)] eV
E_n,j = −(13.6057/n²)[1 + (α²/n²)(n/(j+½) − ¾)] eV
the normal Zeeman splitting
ΔE = μ_B B m_l , μ_B = eℏ/2m_e = 9.2740 × 10⁻²⁴ J T⁻¹ = 0.46686 cm⁻¹ T⁻¹
ΔE = μ_B B m_l , μ_B = eℏ/2m_e = 9.2740 × 10⁻²⁴ J T⁻¹ = 0.46686 cm⁻¹ T⁻¹
the anomalous Zeeman splitting and the Landé g-factor
ΔE = g_J μ_B B m_j , g_J = 1 + [j(j+1) + s(s+1) − l(l+1)] / [2j(j+1)]
ΔE = g_J μ_B B m_j , g_J = 1 + [j(j+1) + s(s+1) − l(l+1)] / [2j(j+1)]
the helium Hamiltonian — and the term that makes it insoluble
Ĥ = −(ℏ²/2m)(∇²_1 + ∇²_2) − Ze²/4πε_0r_1 − Ze²/4πε_0r_2 + e²/4πε_0r_12
Ĥ = −(ℏ²/2m)(∇²_1 + ∇²_2) − Ze²/4πε_0r_1 − Ze²/4πε_0r_2 + e²/4πε_0r_12
Definitions worth memorising
The hydrogen-atom problem: find the bound stationary states of a single electron of charge −e and mass m_e interacting with a single nucleus of charge +Ze and mass M through the Coulomb potential alone, treating both particles non-relativistically and ignoring spin. Z = 1 gives hydrogen itself; Z = 2, 3, … give the one-electron ions He⁺, Li²⁺, and so on.
Reduced mass, μ: the effective mass of the relative motion of two bodies, μ = m_1m_2/(m_1 + m_2). It is always smaller than either mass, and when one body is far heavier than the other it tends to the mass of the lighter one.
Central potential: a potential energy function that depends only on the distance r from a fixed centre, V = V(r), not on the angles. Its defining property is spherical symmetry: rotating the system about the centre leaves the Hamiltonian unchanged. Consequently L̂² and L̂_z both commute with Ĥ, and the wavefunction can be written as a product of a radial and an angular function.
Effective potential: V_eff(r) = V(r) + ℏ²l(l+1)/(2μr²). The second term is the centrifugal barrier: the energy cost of having angular momentum while being close to the centre. It is repulsive, it grows as l(l+1), and it is what keeps p, d and f electrons away from the nucleus.
Accidental degeneracy: a degeneracy not explained by the obvious geometric symmetry of the Hamiltonian. In the hydrogen atom, states of the same n but different l are accidentally degenerate. The name is historical and slightly unfair: the degeneracy is in fact required by a hidden symmetry, but not by a spatial one.
Orbital: a one-electron spatial wavefunction. In hydrogen it is an exact solution ψ_nlm; in every other atom it is an approximation, the one-electron function out of which an approximate many-electron wavefunction is built (Part 6). The word carries no implication of an orbit, a path or a fixed radius.
Radial node: a value of r, strictly between 0 and ∞, at which R_nl(r) = 0 and changes sign. Geometrically it is a spherical surface on which the probability density vanishes. The number of them is n − l − 1.
Angular node: a surface passing through the nucleus — a plane, or in the d_z² case a cone — on which the angular function vanishes and changes sign. There are exactly l of them, for every orbital.
Radial distribution function, P(r): the probability, per unit radial distance, of finding the electron at distance r from the nucleus in any direction. P(r)dr is the probability that the electron lies in the thin spherical shell between r and r + dr. For any orbital, P(r) = r²[R_nl(r)]², and for an s orbital, where |ψ|² is the same in every direction, this can be written 4πr²|ψ|².
Virial theorem (quantum, stationary states): for a potential that is a homogeneous function of the coordinates of degree k, so that V(λr) = λ^kV(r), the expectation values in any stationary state satisfy 2⟨T⟩ = k⟨V⟩.
Where these come from
This sheet is distilled from Quantum Chemistry, Part 5 — 10 sections that derive every one of these results and show you how to use them.
Read Part 5 All formula sheets