Lecture 2 Hamiltonian operators for molecules

1 Lecture 2 Hamiltonian operators for SkylarisCHEM6085: Density Functional TheoryCHEM6085 Density Functional TheoryThe (time-independent) Schr dinger equation is an eigenvalueequationoperator for property AeigenfunctioneigenvalueEnergy operator ( Hamiltonian )wavefunctionEnergy eigenvalue2 CHEM6085 Density Functional TheoryConstructing operators in Quantum MechanicsClassical quantityQuantum operatorQuantum mechanical operators are the same as their corresponding classical mechanical quantitiespositionPotential energy ( energy of attraction of an electron by an atomic nucleus)With one exception!The momentum operator is completely different:3 CHEM6085 Density Functional TheoryBuilding HamiltoniansTheHamiltonianoperator(=tota lenergyoperator)isasumoftwooperators:the kineticenergyoperatorandthepotentialener gyoperatorKineticenergyrequirestakingint oaccountthemomentumoperatorThe potential energy operator is straightforward4 The Hamiltonian becomes:CHEM6085 Density Functional TheoryTheaboveexampleprovidestheexpectat ionvalue(averagevalue)ofthepositionalong thex-axis.

Lecture 2 Hamiltonian operators for molecules C.-K. Skylaris CHEM6085: Density Functional Theory CHEM6085 Density Functional Theory. The (time-independent) Schrödinger equation is an eigenvalue equation operator for property A eigenfunction eigenvalue Energy …

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