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(aver)

the electronic Hamiltonian operator of any molecule, with any number of nuclei and electrons. 2) Write down an expression for the expectation value of each of the terms of the above Hamiltonian (i.e. Kinetic energy, electron-electron repulsion energy, etc.)

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