Computational Chemistry in Quantum Chemistry Studies
Explore the world of computational chemistry in quantum chemistry studies, covering topics like optimizing structures, basis set names, molecular orbitals, Hamiltonian principles, approximations, linear combinations of atomic orbitals, and the variational principle for finding optimal electronic structures.
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Chem 516: Day 20 Computational Chemistry 1
Goals for today Understand what people mean when they say I optimized the structure using B3LYP/6-31G* Learn the difference between Hartree-Fock and Density Functional Theory Know how to read a basis set name and why you care Basically know enough to be able to learn more on your own 2
Group activity In groups: List some things you might want to calculate about a molecule, or that you have read about people calculating 3
Using O2 as an example of molecular orbitals In freshman chemistry, we all learned how to draw pictures like this: But how to do you tell a computer to do that? 4
As usual, lets look at the Hamiltonian 2 2?? Electron kinetic energy H = ??2 ?=????????? 2 2????2 Nuclear kinetic energy + ?=?????? ???2 ??? Electron-nuclear attraction ?? ?2 ??? ?????2 ??? + ? ?>? Electron-electron repulsion + ? ?>? Nuclear-nuclear repulsion 5
The big problem For more than one electron, this is impossible to solve exactly. Quantum chemistry is therefore all about approximations The most common one is the Born-Oppenheimer Approximation: electrons are fast and nuclei are slow Let electrons feel the nuclei as fixed point charges, and then let nuclei feel electrons as just charge clouds. 6
Next approximation: MOs are linear combinations of atomic orbitals (LCAOs) 2 parts to this: 1) Define orbitals: 1 1??+ 1?? 1?= 2 1 1?? 1?? 1? = 2 2) Fill in the electrons: ( 1s)2( 1s*)2 ( x*)1( x*)1 We have choices in both places: I could say that: 1 3 41?? 41??+ 1?= and then fill in electrons Or I could use the original MOs but say that two electrons go in x* 7
How do we know what is the best electronic structure? Variational Principle says that E( best) E( trial) So I can just keep trying different electron configurations until I converge on the best energy I can find. That will be the optimal wavefunction for my molecule best subject to all of the approximations I have made So our task is to start with some educated guess for the molecular orbitals and occupations, then tweak that guess bit by bit. 8
Hartree-Fock theory is one way to keep track of all these electrons Basic idea: calculate the energy of each electron one at a time. That electron feels the average charge of all the other ones: 2 2???? ???2 ???+ ???? 2 ? ? ? = ? = ?? ? 9
Heres the workflow Guess initial MOs (coefficients, AOs) Change those coefficients a little bit Calculate VHF for all e-s No Is the energy difference from the last round 0? Calculate E Yes Done! 10
Density Functional Theory Here s the problem: a molecule like a porphyrin has ~100 electrons. Each has x,y,z coordinates, so that is now 300 variables I need to keep track of. It s more efficient to just keep track of the total electron density than to keep track of each individual electron But now the Hamiltonian isn t so straightforward: H = T + V + U e--e- interaction energy Kinetic Energy Potential from nuclei 11
The problem with DFT The true equation for T and U as a function of the electron density is not known! We have only our best guesses These equations for T and U have names like BP86 , B3LYP , M06 . In general these are the initials of the people who developed them. Becke 3-parameter, Lee-Yang-Parr For most molecules, B3LYP is a good first choice it may not be the most accurate but it is unlikely to give a pathologically bad result 12
Basis Sets But let s go back: what do the AOs we started with even look like? From the solution to the Schroedinger equation for an H atom, 1? = ? ??0: 13
The problem is that you end up needing to calculate lots of overlap integrals There s no analytical equation for this 14
But there are other functions that do have nice overlaps, such as Gaussians: 2 ? ? = ? 1 ? 2 15
But theres not enough electron density at the nucleus: 16
The solution is to add up skinny and wide gaussians Slater Gaussian 1 Gaussian 2 Gaussian 3 Gaussian 3 Sum of gaussians Slater Gaussian 1 Gaussian 2 17
Basis sets can be described as a sum of gaussians 6-31G Core electrons are sum of 6 gaussians Valence electrons have more options: c1* + c2* split-valence, double-zeta
Basis sets can be described as a sum of gaussians 6-311G Core electrons are sum of 6 gaussians Valence electrons have more options: c1* + c3* + c2* split-valence, triple-zeta
Polarization functions Fe Ni 6-31G H Fe Ni 6-31G** H p orbs on H d orbs on C,N,etc 20
A typical input file for a calculation (in this case a program called Gaussian) %Mem=2GB # B3LYP/6-31G* Opt Freq This is my comment describing the molecule so I don t forget what I m doing later Charge, Spin 0,1 C 1.0 0.0 -1.0 N 2.3 1.2 -0.5 Atom coordinates 21
