Quantum Chemistry: Postulates, Model Systems, the Hydrogen Atom and Approximation Methods

The state of a system is described completely by a wave function ψ(r, t). By the Born interpretation, ψ itself has no direct physical meaning but |ψ|² = ψ*ψ is a probability density: |ψ(x)|² dx is the probability of finding the particle between x and x + dx. So an acceptable ψ must be single-valued, continuous, finite and square-integrable, and is normalised so that ∫|ψ|² dτ = 1. In Dirac notation a state is a ket |ψ⟩, its conjugate a bra ⟨ψ|, and ⟨φ|ψ⟩ = ∫φ*ψ dτ is their overlap; normalisation is ⟨ψ|ψ⟩ = 1 and orthogonality ⟨ψₘ|ψₙ⟩ = 0 for m ≠ n.

Every observable corresponds to a linear Hermitian operator: position x̂ = x, momentum p̂ₓ = −iħ ∂/∂x, kinetic energy T̂ = −(ħ²/2m)∇², and the Hamiltonian Ĥ = T̂ + V̂. Hermitian operators have real eigenvalues and orthogonal eigenfunctions for distinct eigenvalues. A measurement can only return an eigenvalue of the operator; for a state that is not an eigenfunction, the average over many measurements is the expectation value ⟨A⟩ = ⟨ψ|Â|ψ⟩/⟨ψ|ψ⟩.

The commutator [Â, B̂] = ÂB̂ − B̂Â decides whether two observables can be known together. The canonical one is [x̂, p̂ₓ] = iħ, which leads to the uncertainty relation Δx·Δpₓ ≥ ħ/2. Operators that commute share a complete set of eigenfunctions, so their observables have simultaneously sharp values: for the hydrogen atom Ĥ, L̂² and L̂_z all commute, which is why n, l and mₗ label one orbital together. The time-dependent Schrödinger equation iħ ∂Ψ/∂t = ĤΨ has, for a time-independent potential, separable solutions Ψ(x, t) = ψ(x)e^(−iEt/ħ), where ψ obeys the time-independent equation Ĥψ = Eψ. These are stationary states: |Ψ|² does not change with time.

For a particle of mass m confined to 0 ≤ x ≤ L by infinite walls, ψ must vanish at both walls. The solutions are ψₙ = √(2/L) sin(nπx/L) with Eₙ = n²h²/(8mL²), n = 1, 2, 3, … The state n has n − 1 interior nodes; the lowest energy is not zero — the zero-point energy h²/(8mL²) is required by the uncertainty principle — and the spacing Eₙ₊₁ − Eₙ = (2n + 1)h²/(8mL²) grows with n and shrinks as the box or the mass grows, which is why a macroscopic object looks classical. In the ground state ⟨x⟩ = L/2 and ⟨pₓ⟩ = 0, while ⟨pₓ²⟩ = h²/(4L²).

In two and three dimensions the Hamiltonian separates: for a rectangular box E = (h²/8m)(nₓ²/a² + n_y²/b² + n_z²/c²). In a square or cubic box different sets of quantum numbers give the same energy — degeneracy that comes from symmetry. In a cube, with ε = h²/(8mL²), the levels are 3ε (1,1,1, non-degenerate), 6ε (three-fold), 9ε (three-fold), 11ε (three-fold), 12ε (2,2,2, non-degenerate) and 14ε (the six permutations of 1,2,3). Stretching one side lifts the degeneracy.

With finite walls of height V₀ the wave function no longer stops at the wall: inside the barrier, where E < V₀, it decays as e^(−κx) with κ = √(2m(V₀ − E))/ħ. There are only a finite number of bound states, each lower in energy than its infinite-well counterpart because the particle has more room. Through a barrier of width a the transmission probability for κa ≫ 1 is roughly T ≈ e^(−2κa): tunnelling falls exponentially with the barrier width and with √m, so it matters for electrons and protons and hardly at all for heavier groups — the basis of the scanning tunnelling microscope, of proton-transfer isotope effects and of the ammonia inversion.

Application — the free-electron model of a conjugated polyene. Treat the N π electrons as particles in a one-dimensional box of length L, two per level. The HOMO is n = N/2 and the LUMO n = N/2 + 1, so the lowest absorption is ΔE = (N + 1)h²/(8mₑL²) and λ = 8mₑL²c/[(N + 1)h]. Lengthening the chain raises L faster than N + 1, so λ moves to longer wavelength — the red shift of longer polyenes and carotenoids.

For V = ½kx² the energies are E_v = (v + ½)hν, v = 0, 1, 2, …, with ν = (1/2π)√(k/μ) and μ the reduced mass m₁m₂/(m₁ + m₂) for a diatomic. The levels are equally spaced by hν and the ground state keeps the zero-point energy ½hν. The wave functions are ψ_v = N_v H_v(y) e^(−y²/2), with y = x√α, α = √(kμ)/ħ, and N_v = (α/π)^¼ (2^v v!)^(−½). The Hermite polynomials are H₀ = 1, H₁ = 2y, H₂ = 4y² − 2, H₃ = 8y³ − 12y, generated by the recursion H_(v+1) = 2yH_v − 2vH_(v−1). Each H_v has v real roots, so ψ_v has v nodes, and parity (−1)^v: even v gives an even function, odd v an odd one — which is what makes ⟨x⟩ = 0 in every state and ⟨v|x|v′⟩ non-zero only for v′ = v ± 1.

For the harmonic potential the virial theorem gives ⟨T⟩ = ⟨V⟩ = E_v/2 in every stationary state. Classically the oscillator never passes its turning points, but ψ₀ extends beyond them: about 16% of the ground-state probability lies in the classically forbidden region. Real bonds are anharmonic — the Morse potential V = Dₑ(1 − e^(−a(r−rₑ)))² is the usual model — and the levels become G(v) = (v + ½)ω̃ₑ − (v + ½)²ω̃ₑxₑ, converging as v rises. The spectroscopic dissociation energy D₀ is measured from v = 0 and is smaller than the well depth Dₑ by the zero-point energy.

The orbital angular momentum operators obey [L̂ₓ, L̂_y] = iħL̂_z and cyclic permutations, so no two components can be sharp together; but L̂² commutes with each component, and the conventional choice is to make L̂² and L̂_z sharp. Their common eigenfunctions are the spherical harmonics Y_l^m(θ, φ) = Θ_lm(θ)e^(imφ), with L̂²Y = l(l + 1)ħ²Y and L̂_zY = mħY, l = 0, 1, 2, … and m = −l, …, +l. The Y_l^m are orthonormal over the sphere, have l angular nodes (nodal planes or cones) and parity (−1)^l. Spin is an intrinsic angular momentum with the same algebra: for an electron s = ½, S² = s(s + 1)ħ² = ¾ħ² and m_s = ±½.

A rigid rotor (a diatomic with fixed bond length r and moment of inertia I = μr²) has energies E_J = J(J + 1)ħ²/(2I), J = 0, 1, 2, …, each (2J + 1)-fold degenerate. In spectroscopic units F(J) = BJ(J + 1) with the rotational constant B = h/(8π²cI) in cm⁻¹, so adjacent levels are 2B(J + 1) apart. A particle confined to a ring (rotation in a plane) has E = m²ħ²/(2I), m = 0, ±1, ±2, …: every level above m = 0 is doubly degenerate, clockwise and anticlockwise.

For one electron bound to a nucleus of charge Ze the Schrödinger equation separates into ψ_nlm = R_nl(r)Y_l^m(θ, φ). The energy depends only on n: Eₙ = −Z²Eₕ/(2n²) = −13.6 Z²/n² eV, so a level n is n²-fold degenerate (2n² with spin). The orbital ψ_nlm has n − 1 nodes in all: n − l − 1 radial and l angular. The radial distribution function P(r) = r²R_nl² gives the probability of finding the electron in a shell between r and r + dr; for 1s, P(r) ∝ r²e^(−2Zr/a₀) peaks at the most probable radius a₀/Z, while the mean radius is ⟨r⟩ = 3a₀/(2Z). In general ⟨r⟩ = (a₀/2Z)[3n² − l(l + 1)].

Atomic units set ħ = mₑ = e = 4πε₀ = 1. Lengths are then in bohr (a₀ = 52.92 pm) and energies in hartree (Eₕ = 27.211 eV), so the hydrogen ground state is −½ Eₕ and the Hamiltonian of hydrogen is simply −½∇² − 1/r. The angular distribution is |Y_l^m|²: s orbitals are spherical, the p_z orbital ∝ cos θ has a nodal plane at θ = 90°, and real p and d orbitals are linear combinations of the complex Y_l^(±m).

With two or more electrons the repulsion terms e²/(4πε₀r_ij) prevent separation, so the orbital approximation writes the wave function as a product of one-electron orbitals, each electron moving in the average field of the rest (the idea behind the Hartree–Fock method). Electrons are fermions, so the total wave function must be antisymmetric under exchange of any two electrons, space and spin together — the general form of the Pauli exclusion principle. A Slater determinant builds this in: for helium 1s², Ψ = (1/√2)|1sα(1) 1sβ(1); 1sα(2) 1sβ(2)|. Swapping two electrons swaps two rows and changes the sign; putting two electrons in the same spin-orbital makes two columns equal and the determinant vanish.

The variational principle states that for any well-behaved trial function φ, E[φ] = ⟨φ|Ĥ|φ⟩/⟨φ|φ⟩ ≥ E₀, the true ground-state energy. Minimising E with respect to parameters in φ gives the best upper bound of that form. For helium with the trial function a product of two 1s orbitals of effective charge ζ, E(ζ) = ζ² − 27ζ/8 hartree, minimised at ζ = 27/16 = 1.6875, giving E = −(27/16)² = −2.848 Eₕ (−77.5 eV) against the experimental −2.904 Eₕ (−79.0 eV). The effective charge below 2 is the screening of each electron by the other. In a linear variation φ = Σcᵢχᵢ the minimisation leads to the secular equations Σcᵢ(Hᵢⱼ − ESᵢⱼ) = 0, which have non-trivial solutions only when the secular determinant |Hᵢⱼ − ESᵢⱼ| = 0; its roots are upper bounds to successive states.

First-order non-degenerate perturbation theory splits Ĥ = Ĥ⁽⁰⁾ + Ĥ′ with Ĥ⁽⁰⁾ exactly solvable. The first-order energy correction is the expectation value of the perturbation in the unperturbed state, E⁽¹⁾ = ⟨ψ⁽⁰⁾|Ĥ′|ψ⁽⁰⁾⟩, and the first-order wave function mixes in other states with coefficients ⟨ψₖ⁽⁰⁾|Ĥ′|ψₙ⁽⁰⁾⟩/(Eₙ⁽⁰⁾ − Eₖ⁽⁰⁾). The second-order correction to the ground state is always negative. For helium, taking the electron repulsion as Ĥ′, E⁽⁰⁾ = −108.8 eV (two hydrogen-like electrons with Z = 2) and E⁽¹⁾ = (5/8)Z Eₕ = +34.0 eV, giving −74.8 eV — a poorer estimate than the variational −77.5 eV.