Computational Chemistry at TCU Benjamin G. Janesko TCU 2010.02.19 - PowerPoint PPT Presentation

Computational Chemistry at TCU Benjamin G. Janesko TCU 2010.02.19 Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 1 / 45

Outline Computational Chemistry 1 Drug design 2 Visualizing molecules 3 Quantum chemistry and computing 4 Applying computational chemistry 5 Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 2 / 45

What is computational chemistry? Predicting how atoms, molecules, and solids will behave in new situations Designing new molecules and solids to do new jobs Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 3 / 45

Three areas where computational chemistry is useful Designing drug candidates to cure diseases ▶ Need to predict how individual molecules will stick together Designing molecules with new properties ▶ High-energy-density materials (explosives) Designing catalysts to make products (fuel, plastics, etc) using renewable sources, with less energy ▶ Need to predict how chemical bonds are formed and broken Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 4 / 45

Outline Computational Chemistry 1 Drug design 2 Visualizing molecules 3 Quantum chemistry and computing 4 Applying computational chemistry 5 Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 5 / 45

Designing drugs with computational chemistry http://www.rcsb.org/pdb/explore/jmol.do?structureId=3K2P&bionumber=1 Crystal structure of a drug molecule bound to HIV-1 reverse transcriptase [View in web browser with Jmol applet] Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 6 / 45

How do drugs work? Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 7 / 45

How do drugs work? Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 8 / 45

Drug docking in computational chemistry Determine the structure of a protein target to block Guess several (10 5 ) structures of possible drugs Simulate the binding between drug and protein with ball-and-spring molecular mechanics Take the 5 − 10 best candidates and test in the lab Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 9 / 45

A simple example of drug docking Met-enkephalin pentapeptide structure from http://www.rcsb.org/pdb/explore/explore.do?structureId=1PLX ▶ Endogeneous opioid peptide neurotransmitter ▶ Blocking enkephalins could change pain tolerances Molecular mechanics simulation with ”Avogadro” open-source molecular mechanics package, http://avogadro.openmolecules.net/wiki/Main Page Real-time ”docking” simulation possible [Illustrate drug docking onto met-enkephalin with Avogadro] Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 10 / 45

Outline Computational Chemistry 1 Drug design 2 Visualizing molecules 3 Quantum chemistry and computing 4 Applying computational chemistry 5 Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 11 / 45

Chemists visualize molecules with stereo views . . . Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 12 / 45

. . . or with molecular models . . . Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 13 / 45

. . . and most recently with computers Software like Avogadro could be a valuable classroom tool Build chemical intuition into something you can’t see Electron density maps, charge distributions, etc. are all useful Sometimes it’s enough just to see the molecule in 3-D Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 14 / 45

Example: Adamantane, and other diamandoid hydrocarbons [view in Avogadro] Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 15 / 45

Another fun example: Octanitrocubane [view in Avogadro] Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 16 / 45

Another fun example: Octanitrocubane C 8 (NO 2 ) 8 Decomposes into 8 CO 2 and 4 N 2 This decomposition releases an enormous amount of energy, as your students can see from a simple bond energy calculation Count Bond DE (kJ/mol) 12 C-C 348 8 C-N 308 16 N-O 201 16 C=O 805 4 N[triple]N 941 Total energy change is 9856 - 16644 = 6788 kJ/mol Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 17 / 45

Synthesis of octanitrocubane Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 18 / 45

Outline Computational Chemistry 1 Drug design 2 Visualizing molecules 3 Quantum chemistry and computing 4 Applying computational chemistry 5 Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 19 / 45

Molecular mechanics Force on atom i at position ⃗ r i is Bound to i close to i ∑ r j ∣ − r 0 ∑ − T ijk ( 휃 ijk − 휃 0 ( ) = − k ij ∣ ⃗ r i − ⃗ + F i ij ijk j jk all atoms ⃗ r i − ⃗ r j ∑ + Z i Z j r j ∣ 3 + . . . ∣ ⃗ r i − ⃗ j Atom i accelerates via Newton’s equation F i = m i a i Let each atom move at its new velocity for a short time Δ t , then re-evaluate all the forces Repeat until the atoms have stopped, and you’re at an energy minimum Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 20 / 45

Ball-and-spring models Molecular mechanics is fine for normal molecules . . . Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 21 / 45

Ball-and-spring models Molecular mechanics is fine for normal molecules . . . Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 21 / 45

Quantum mechanical models Electrons in a molecule really obey Schr¨ odinger’s equation from quantum mechanics i ℏ ∂ ˆ ∂ t Ψ( ⃗ x 1 ,⃗ x 2 . . .⃗ x N ) = H Ψ( ⃗ x 1 ,⃗ x 2 . . .⃗ x N ) Closed-form (paper and pencil) solutions to Schr¨ odinger’s equation are available for the hydrogen atom Every other atom, molecule, and solid must be approximated! Those approximations can readily be evaluated by a computer Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 22 / 45

Approximating the wave function of one electron Stationary states of Schr¨ odinger’s equation become − ℏ 2 ∇ 2 휙 i ( ⃗ x ) + V ( ⃗ x ) 휙 i ( ⃗ x ) = 휖 i 휙 i ( ⃗ x ) 2 m e Everything we need to know about that electron can be obtained from its wave function We can approximate the wavefunction as a sum of “basis functions” M ∑ 휙 i ( ⃗ x ) = c i 휇 휒 휇 ( ⃗ x ) 휇 =1 Select the basis functions so that the math becomes easy Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 23 / 45

A basis set for the electron in H + 2 1.0 Basis function amplitude 0.8 0.6 0.4 0.2 0.0 � 2 � 1 0 1 2 3 Z � coordinate ) 1 / 4 ( 2 훼 ( R 휇 ∣ 2 ) r − ⃗ 휒 휇 ( ⃗ r ) = − 훼 ∣ ⃗ exp 휋 Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 24 / 45

Approximating the wave function of one electron Describe the wavefunction as a vector ⃗ 휙 . Element 휇 of the vector is the coefficient c i 휇 in M ∑ 휙 i ( ⃗ r ) = c i 휇 휒 휇 ( ⃗ r ) 휇 =1 Substituting that expression into Schr¨ odinger’s equation − ℏ 2 ∇ 2 휙 i ( ⃗ r ) + V ( ⃗ r ) 휙 i ( ⃗ r ) = 휖 i 휙 i ( ⃗ r ) 2 m e gives a new matrix equation ∑ = H 휇휈 휙 휈 휖휙 휇 휈 − ℏ 2 ∫ [ ] 2 m ∇ 2 + V ( ⃗ d 3 ⃗ H 휇휈 = r 휒 ∗ 휇 ( ⃗ r ) r ) 휒 휈 ( ⃗ r ) The coefficients c i 휇 , which are all we need to describe the wave function, are eigenvectors of the matrix H Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 25 / 45

Computers are great for diagonalizing matrices BLAS/LAPACK (Basic Linear Algebra Subprograms / Linear Algebra PACKage) suite of matrix diagonalization algoritms in FORTRAN/C/C++ Jacobi eigenvalue algorithm, Cholesky decomposition, LU factorization . . . Algorithms specific to sparse matrices (most elements are zero) Quantum chemistry codes are really just front-ends to BLAS/LAPACK! Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 26 / 45

Our H + 2 example 2 basis functions, one on each atom H is a 2 x 2 matrix 휓 is a 2-element vector that obeys ( 훼 ) ( c 1 ( c 1 ) ) 훽 = 휖 훽 훼 c 2 c 2 The 2 eigenvalues of H correspond to occupied and unoccupied orbitals Realistic calculations will use hundreds or thousands of basis functions Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 27 / 45

Approximating the wavefunction of many electrons Approximate as a product of one-electron functions Ψ( ⃗ r 1 ,⃗ r 2 . . .⃗ r n ) = 휙 1 ( ⃗ r 1 ) 휙 2 ( ⃗ r 2 ) . . . 휙 N ( ⃗ r N ) ▶ Detail: Add ”exchange” terms to ensure Ψ changes sign when 2 electrons are interchanged Each electron obeys − ℏ 2 ∇ 2 휙 i ( ⃗ r ) + V ( ⃗ r ) 휙 i ( ⃗ r ) = 휖 i 휙 i ( ⃗ r ) 2 m e where V ( ⃗ r ) is the interaction with the average electron distribution ▶ Nonlocal ”exchange” contributions avoid self-interaction Now we occupy the N lowest-energy eigenvectors of H Iterate to self-consistency Benjamin G. Janesko (TCU) Computational Chemistry at TCU 2010.02.19 28 / 45