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The Schrödinger Equation

Typically referring to the time-independent Schrodinger Equation:

H^Ψ=EΨ\hat{H}\Psi = E \Psi

which relates the total energy of the system, EE, its wavefunction Ψ\Psi, a function of particle coordinates (spatial or momentum, spin), and the Hamiltonian operator H^\hat{H}.

The Hamiltonian for a typical chemical system will be something like:

H^=T^e+V^ee+V^en+T^n+V^nn\hat{H} = \hat{T}_e + \hat{V}_{ee} + \hat{V}_{en} + \hat{T}_n + \hat{V}_{nn}

where T^e\hat{T}_e and T^n\hat{T}_n are kinetic energy terms for the electrons ee and nuclei nn and V^ee\hat{V}_{ee}, V^en\hat{V}_{en} and V^nn\hat{V}_{nn} are potential energy terms for electron-electron, electron-nuclear and nuclear-nuclear interactions.

T^e=−∇2i2\hat{T}_e = - \frac{\nabla^2 i}{2}
T^n=−∇2A2MA\hat{T}_n = - \frac{\nabla^2 A}{2 M_A}
V^ee=∑i<j1rij\hat{V}_{ee} = \sum_{i<j} \frac{1}{r_{ij}}
V^en=−∑A,iZArAi\hat{V}_{en} = -\sum_{A,i} \frac{Z_A}{r_{Ai}}
V^nn=∑A<BZAZBrAB\hat{V}_{nn} = \sum_{A<B} \frac{Z_A Z_B}{r_{AB}}

where:

  • ii and jj indicate electrons,
  • AA and BB indicate nuclei with nuclear charge ZZ and mass MM,
  • ∇\nabla is the Laplacian