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Ch.1. Elementary Quantum Chemistry. References (on-line) A brief review of elementary quantum chemistry http://zopyros.ccqc.uga.edu/lec_top/quantrev.bak Wikipedia ( http://en.wikipedia.org ) Search for Schrodinger equation, etc.
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Ch.1. Elementary Quantum Chemistry • References (on-line) • A brief review of elementary quantum chemistry • http://zopyros.ccqc.uga.edu/lec_top/quantrev.bak • Wikipedia (http://en.wikipedia.org) • Search for Schrodinger equation, etc. • Molecular Electronic Structure Lecturewww.chm.bris.ac.uk/pt/harvey/elstruct/introduction.html
The Schrödinger equation The ultimate goal of most quantum chemistry approach is the solution of the time-independent Schrödinger equation. (1-dim) Hamiltonian operator wavefunction (solving a partial differential equation)
Postulate #1 of quantum mechanics • The state of a quantum mechanical system is completely specified by the wavefunction or state function that depends on the coordinates of the particle(s) and on time. • The probability to find the particle in the volume element located at r at time t is given by . (Born interpretation) • The wavefunction must be single-valued, continuous, finite, and normalized (the probability of find it somewhere is 1). • = <|> Probability density
Postulate #2 of quantum mechanics • Once is known, all properties of the system can be obtained • by applying the corresponding operators to the wavefunction. • Observed in measurements are only the eigenvalues a which satisfy • the eigenvalue equation Schrödinger equation: Hamiltonian operator energy with (Hamiltonian operator) (e.g. with )
Postulate #3 of quantum mechanics • Although measurements must always yield an eigenvalue, • the state does not have to be an eigenstate. • An arbitrary state can be expanded in the complete set of • eigenvectors ( as where n . • We know that the measurement will yield one of the values ai, but • we don't know which one. However, • we do know the probability that eigenvalue ai will occur ( ). • For a system in a state described by a normalized wavefunction , • the average value of the observable corresponding to is given by • = <|A|>
Atomic units (a.u.) • Simplifies the Schrödinger equation (drops all the constants) • (energy) 1 a.u. = 1 hartree = 27.211 eV = 627.51 kcal/mol, • (length) 1 a.u. = 1 bohr = 0.52918 Å, • (mass) 1 a.u. = electron rest mass, • (charge) 1 a.u. = elementary charge, etc. (before) (after)
Born-Oppenheimer approximation • Simplifies further the Schrödinger equation (separation of variables) • Nuclei are much heavier and slower than electrons. • Electrons can be treated as moving in the field of fixed nuclei. • A full Schrödinger equation can be separated into two: • Motion of electron around the nucleus • Atom as a whole through the space • Focus on the electronic Schrödinger equation
Born-Oppenheimer approximation (before) (electronic) (nuclear) E = (after)
Antisymmetry and Pauli’s exclusion principle • Electrons are indistinguishable. Probability doesn’t change. • Electrons are fermion (spin ½). antisymmetric wavefunction • No two electrons can occupy the same state (space & spin).
= = Variational principle • Nuclei positions/charges & number of electrons in the molecule • Set up the Hamiltonial operator • Solve the Schrödinger equation for wavefunction , but how? • Once is known, properties are obtained by applying operators • No exact solution of the Schrödinger eq for atoms/molecules (>H) • Any guessed trial is an upper bound to the true ground state E. • Minimize the functional E[] by searching through all acceptable • N-electron wavefunctions
but not antisymmetric! Many-electron wavefunction: Slater determinant • Impossible to search through all acceptable N-electron wavefunctions • Let’s define a suitable subset. • N-electron wavefunctionaprroximated by a product ofN one-electron • wavefunctions(hartree product) • It should be antisymmetrized ( ).
= 0 Slater “determinants” • A determinant changes sign when two rows (or columns) are exchanged. • Exchanging two electrons leads to a change in sign of the wavefunction. • A determinant with two identical rows (or columns) is equal to zero. • No two electrons can occupy the same state. “Pauli’s exclusion principle” “antisymmetric” = 0
= ij Hartree-Fock (HF) approximation • Restrict the search for the minimum E[] to a subset of , which • is all antisymmetric products of N spin orbitals (Slater determinant) • Use the variational principle to find the best Slater determinant • (which yields the lowest energy) by varying spin orbitals (orthonormal)
Slater determinant (spin orbital = spatial orbital * spin) Hartree-Fock (HF) energy: Evaluation finite “basis set” Molecular Orbitals as linear combinations of Atomic Orbitals (LCAO-MO) where
where Hartree-Fock (HF) equation: Evaluation • No-electron contribution (nucleus-nucleus repulsion: just a constant) • One-electron operator h (depends only on the coordinates of one electron) • Two-electron contribution (depends on the coordinates of two electrons)
Potential energy due to nuclear-nuclear Coulombic repulsion (VNN) *In some textbooks ESD doesn’t include VNN, which will be added later (Vtot = ESD + VNN). • Electronic kinetic energy (Te) • Potential energy due to nuclear-electronic Coulombic attraction (VNe)
Potential energy due to two-electron interactions (Vee) • Coulomb integral Jij (local) • Coulombic repulsion between electron 1 in orbital i and electron 2 in orbital j • Exchange integral Kij (non-local) only for electrons of like spins • No immediate classical interpretation; entirely due to antisymmetry of fermions > 0, i.e., a destabilization
Self-Interaction • Coulomb term J when i = j (Coulomb interaction with oneself) • Beautifully cancelled by exchange term K in HF scheme 0 = 0
and Hartree-Fock (HF) equation • Fock operator: “effective” one-electron operator • two-electron repulsion operator (1/rij) replaced by one-electron operator VHF(i) • by taking it into account in “average” way Two-electron repulsion cannot be separated exactly into one-electron terms. By imposing the separability, the Molecular Orbital Approximation inevitably involves an incorrect treatment of the way in which the electrons interact with each other.
Self-Consistent Field (HF-SCF) Method • Fock operator depends on the solution. • HF is not a regular eigenvalue problem that can be solved in a closed form. • Start with a guessed set of orbitals; • Solve HF equation; • Use the resulting new set of orbitals • in the next iteration; and so on; • Until the input and output orbitals • differ by less than a preset threshold • (i.e. converged).
Koopman’s theorem • As well as the total energy, one also obtains a set of orbital energies. • Remove an electron from occupied orbital a. Orbital energy = Approximate ionization energy
Electron correlation • A single Slater determinant never corresponds to the exact wavefunction. • EHF > E0 (the exact ground state energy) • Correlation energy: a measure of error introduced through the HF scheme • EC = E0- EHF (< 0) • Dynamical correlation • Non-dynamical (static) correlation • Post-Hartree-Fock method • Møller-Plesset perturbation: MP2, MP4 • Configuration interaction: CISD, QCISD, CCSD, QCISD(T)
