qchemsummary - A Short Summary of Quantum Chemistry Quantum...

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A Short Summary of Quantum Chemistry Quantum Chemistry is (typically) based on the non-relativistic Schroedinger equation, making the Born-Oppenheimer approximation. The Schroedinger equation is H total Ψ total = E Ψ total Where E = an allowed energy of the system (the system is usually a molecule). Ψ total = a function of the positions of all the electrons and nuclei and their spins. H total = a differential operator constructed from the classical Hamiltonian H(p,q) = E by replacing all the momenta p i with (h/2 π i) / q i as long as all the p and q are Cartesian. If you would prefer to use a non-Cartesian coordinate system it is tricky to construct the correct quantum mechanical H. The standard way to make Hamiltonians for arbitrary coordinate systems is to use the “Podolsky trick”. For a system of nuclei and electrons in a vacuum with no external fields, neglecting magnetic interactions, using atomic units: H total = - ½ Σ i 2 /M i - ½ Σ n 2 + Σ Z i Z j /|R i -R j | - Σ Z i /|R i -r n | + Σ 1/|r m -r n | The Born-Oppenheimer approximation is to neglect some of the terms coupling the electrons and nuclei, so one can write: Ψ total (R,r) = Ψ nuclei (R) Ψ electrons (r; R) H total T nuclei (P,R) + H electrons (p,r;R) (ignores the dependence of H electrons on the momenta of the nuclei P) then solve the Schroedinger equation for the electrons (with the nuclei fixed). The energy we compute will depend on the positions R of those fixed nuclei, call it V(R): H electons (p,r;R) Ψ electrons (r; R) = V(R) Ψ electrons (r; R) Now we go back to the total Hamiltonian, and integrate over all the electron positions r, ignoring any inconvenient terms, to obtain an approximate Schroedinger equation for the nuclei: < Ψ electrons (r; R) |H total | Ψ electrons (r; R)> H nuclei = T nuclei (P,R) + V(R) Both approximate Schroedinger equations are still much too hard to solve exactly (they are partial differential equations in 3N particles coordinates), so we have to make more approximations. V nuclei is usually expanded to second order R about a stationary point R o : V nuclei V nuclei (R o ) + ½ Σ ( 2 V/ R i R j ) (R i -R oi )(R j -R oj )
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and then the translations, rotations, and vibrations are each treated separately, neglecting any inconvenient terms that couple the different coordinates. In this famous “rigid-rotor-harmonic-oscillator (RRHO)” approximation, analytical formulas are known for the energy eigenvalues, and for the corresponding partition functions Q, look in any P.Chem. text. This approximate approach has the important advantage that we do not need to solve the Schroedinger equation for the electrons at very many R’s: we just need to find a stationary point R o , and compute the energy and the second derivatives at that R o . Many computer programs have been written that allow one to compute the first and second derivatives of V almost as quickly as you can compute V. For example, for the biggest calculation called for in problem 3 with 10 atoms and
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qchemsummary - A Short Summary of Quantum Chemistry Quantum...

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