Chemistry 6440 / 7440 Model Chemistries Electronic Structure Theory Schrodinger equation HY = EY The many-electron CI expansion Y ( R , r ) C ( R ) ( R , r ) ( R , r ) Mi O ( R , r ) i The one-electron LCAO expansion AO MO ( R , r ) C ( R ) ik i k (r ) k Model Chart Minimal STO-3G HF MP2 MP3 MP4 QCISD(T) ... Full CI Split-Valence 3-21G Basis Polarized 6-31G* 6-311G* Diffuse 6-311+G* High Ang. Mom. 6-311+G(3df,p) … Schrödinger Equation Model Chemistries • A theoretical model chemistry is a complete algorithm for the calculation of the energy of any molecular system. • It cannot involve subjective decisions in its application. • It must be size consistent so that the energy of every molecular species is uniquely defined. • A simple model chemistry employs a single theoretical method and basis set. • A compound model chemistry combines several theoretical methods and basis sets to achieve higher accuracy at lower cost. • A model chemistry is useful if for some class of molecules it is the most accurate calculation we can afford to do. Development of a Model Chemistry • Set targets – accuracy goals – cost/size goals – validation data set • Define and implement methods – Specify level of theory for geometry optimization, electronic energy, vibrational zero point energy • Test model on validation data set Evaluating Model Chemistries (Page 147 in Exploring Chemistry) Largest Errors Model Chemistry B3LYP/6-311+G(2d,p) // B3LYP/6-311+G(2d,p) BLYP/6-311+G(2d,p) // BLYP/6-311+G(2d,p) BLYP/6-31+G(d,p) // BLYP/6-31+G(d,p) B3LYP/6-31+G(d,p) // B3LYP/6-31+G(d,p) B3LYP/6-31G(d) // B3LYP/6-31G(d) MP2/6-311+G(2d,p) // MP2/6-311+G(2d,p) MP2/6-31+G(d,p) // MP2/6-31+G(d,p) PM3 // PM3 SVWN5/6-311+G(2d,p) // SVWN5/6-311+G(2d,p) AM1 // AM1 SVWN/6-311+G(2d,p) // SVWN/6-311+G(2d,p) HF/6-31+G(d,p) // HF/6-31+G(d,p) HF/6-31G(d) // HF/6-31G(d) HF/3-21G(d) // HF/3-21G(d) HF/STO-3G // HF/STO-3G MAD 3.1 3.9 3.9 3.9 7.9 8.9 11.4 17.2 18.1 18.8 24.9 46.7 51.0 58.4 93.3 StdDev 3.0 3.2 3.2 4.2 9.5 7.8 8.1 14.0 19.8 16.9 19.2 40.6 41.2 50.1 66.3 Positive Negative -19.7 13.6 -15.9 14.3 -15.2 15.2 -33.8 17.6 -54.2 12.2 -39.2 18.3 -44.0 15.6 -69.9 49.6 -10.1 81.0 -95.5 47.8 -10.4 89.3 -179.8 10.1 -184.2 11.5 -215.2 19.5 -313.9 101.3 Compound Model Chemistries: G2 and G2(MP2) • Proposed by J. Pople and co-workers • Goal: Atomization energies to 2 kcal/mol • Strategy: Approximate QCISD(T)/6311+G(3df,p) by assuming that basis set and correlation corrections are additive • Mean absolute error of 1.21 kcal/mol in 125 comparisons REQUIRED ACCURACY FOR CHEMISTRY Components of the calculated dissociation energy (mEh) for 02. SCF MP2 MP3,4 QCISD(T) Core ZPE O2 3 g -149,691.91 -496.12 6.92 -16.48 -118.16 4.22 2O 3Po Do -149,637.86 -337.96 -23.14 -6.77 -116.81 0. -30.06 9.70 1.35 -4.22 CPU time N2.4 54.05 158.17 N4 N6 N7 N5 For 1 millihartree accuracy: •Need SCF energy to six figures •MP2 Contribution to three figures •Higher order correlation to two figures •Core and ZPE contribution to one figure *Octerski, Joseph W., G.A. Petersson, and J.A. Montgomery, “A complete basis set model chemistry. V. Extensions to six or more heavy atoms”, J.Chem. Phys., 104,2598, (1996). N3.4 Cost-Effective Models • Small basis set, low-order geometry and ZPE • Large SCF basis set – Medium MP2 basis set – Small MP3, MP4(SDQ), CCSD(T) basis sets • Cancellation of errors in 3-body effects (CBS-4) • CBS2 extrapolation of MP2 energy reduces the need for large MP2 basis set • Size consistent empirical corrections CBS Extrapolation The slow, N-1, convergence of the correlation energy vs the one-electron basis set expansion is the result of the universal cusp in wave functions as interelectronic distances, rij 0 . Thus, we can reasonably expect the N-1 form to also be universal. SCF MP2 MP4(SDQ) MP4(SDTQ) QCISD(T) FCI 6-31G 631G† 6-31+G† 6-31+G†† 6-311G(d,p) 6-31+G(d(f),d,p) 6-311+G(d,p) 6-311G(2df,p) 6-311+G(2df,p) 6-311+G(3df,2p) 6-311+G(3d2f,2df,p) 6-311++G(3d2f,2df,2p) [6s6p3d2f,4s2p1d] CBS Exact RM S Error: G 2 test set (kcal/m ol) 3.0 2.5 C BS-4 C BS-q 2.0 G 2(MP2) G2 1.5 C BS-Q B3 1.0 C BS-Q C I/APNO 0.5 0.0 0 5 10 15 20 25 M axim um N um ber of H eavy Atom s 30 Compound Model Chemistries: Thermochemistry The CBS-QCI/APNO model shows the best performance with this test set of any general theoretical model proposed to date, with a mean absolute deviation (MAD) from experiment of 0.53 kcal/mol. Of the models defined for both the first-and second-row elements, the CBSQB3 model gives the greatest accuracy (MAD=0.87 kcal/mol), followed by G2 theory (MAD=1.21 kcal/mol), G2(MP2) theory (MAD=1.59 kcal/mol), the CBS-q model (MAD=1.71 kcal/mol) and finally, the CBS4 model (MAD=1.98 kcal/mol). All are at or under the accuracy of ~2 kcal/mole required for meaningful thermochemical predictions.