G1-3 These methods are based on ab initio molecular orbital calculations. Electron correlation is calculated using MP2 or MP4 and QCI.

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1 23. High Accuracy Energy Methods 23.1 Gaussian-n Methods The method keywords G1, G2, G2MP2, G3, G3MP2, G3B3, G3MP2B3, G4, and G4MP2 perform high accuracy complex energy computations in Gaussian. Essentially, this means that they use other defined calculations in tandem to compute energies with an error tolerance of < 2 kcal/mol. All of these methods have been most rigorously tested on molecules constructed from row 1 and row 2 atoms of the periodic table (H, He, Li-F, Na-Cl). Each method was developed as an improvement over the one before: G2 is an improvement upon G1, G3 on G2, and G4 on G3. All methods were developed by Larry Curtiss et al. in 1989, 1991, 1998, and 2007 for G1-4 respectively. High accuracy in these methods is obtained by determining (experimentally and theoretically) a higher level correction E(HLC) based on calculations for the hydrogen atom and hydrogen molecule. The basis of all G-n calculations is that the energies from lower level calculations (MP2,MP4) are added to a higher level correction (HLC) calculation (QCISD(T)): all of it (E) equals some of it (MPn calculations) plus the rest of it (HLC). E e (Gn) = E 0 + E(HLC) (159) G1-3 These methods are based on ab initio molecular orbital calculations. Electron correlation is calculated using MP2 or MP4 and QCI. G1 uses three polarization functions: diffuse-sp, double-d, and f-polarization. Electron correlation is handled using Møller-Plesset Perturbation theory (MP4) and quadratic configuration interaction (QCI). The method is high accuracy for: atomization (dissociation), ionization energies, and electron affinities. This method uses MP2(full)/6-31G*. The inclusion of full means that all electrons are considered in energy calculations, not just valence electrons. The final energy is calculated by adding a zero-point energy (ZPE) that has been calculated using HF/6-3lG* and scaled by the scaling factor G2 improves on G1 by: 1. Correcting for the non-additivity of diffuse sp polarization by including a 2df basis set extension 2. Using a basis set that contains a third d function on non-hydrogen atoms and a second p function on hydrogen 3. Modifying the HLC The main improvements over G1 are: 1. More accurate atomization energies row 1 atom-containing molecules like LiF (35% better) 2. More accurate for hypervalent species 3. More accurate electron affinities for row 2 atom-containing molecules (44% better)

2 The increased accuracy for atomization of ionic species such as LiF comes from correcting the duplication of the contribution from the diffuse polarization function on the negative center (F) by the polarization extension of the positive center (Li). For LiF, the dissociation energy changes from kcal/mol in G1 to kcal/mol in G2- a big difference! G2MP2 uses MP2 instead of MP4 in calculations, significantly decreasing computational cost with negligible loss of accuracy. Both G2 and G2MP2 use the 6-311G* basis set. G3 uses 6-31G* instead of 6-311G*, thereby decreasing computational cost dramatically. G3 corrects for spin-orbit coupling in atoms and core electron correlation. Making these changes increases the accuracy of enthalpies of formation. The HLC is split into two parts: one for atoms and one for molecules. Doing this has significant impacts on calculated values for electron affinity and ionization potential. G3 has higher accuracy than G2 and G1 for: 1. Non-hydrogen systems like SiF4 and CF4 2. Substituted hydrocarbons 3. Unsaturated cyclic species The average error in enthalpy of formation is < 1kcal/mol away from experimentally and theoretically determined values. The average error is 1.02 kcal/mol- 31% more accurate in G3 than it was in G2 (1.48 kcal/mol). The process of G3: 1. Initial equilibrium structure is determined at the HF/6-31G(d) level a. RHF for singlets b. UHF for not singlets (same as G2) 2. HF/6-31G(d) used to calculate frequencies. A scaling factor of is applied to get ZPE (same as G2) 3. Equilibrium structure reoptimized at MP2(full) 6-31G(d). All electrons are taken into consideration when calculating correlation energies 4. Single point energy calculations carried out at MP4/6-31G(d). This is then modified by: a. Correction for diffuse function b. Correction for higher polarization function c. Quadratic CI correction by QCISD d. Correction for larger basis set effects caused by assumption of separate basis set extensions for diffuse functions and higher polarization functions Step 4 is what most differentiates G3 from G2. It uses the 6-31G basis set rather than the 6-311G, includes more polarizations on second row and less on first, and core

3 polarization functions are added. To account for core-related correlation contributions to total energy, step 4d is carried out at MP2(full). 5. MP4/6-31G(d), the energy from step 4, and a spin orbit correction are added together for atomic species. Spin orbit correction, determined experimentally or theoretically, is an important consideration for halide containing compounds. Molecular spin orbit interaction is neglected because isgives no overall improvement to energy. 6. HLC is added in. Includes corrections (A, B, C, and D) for pairs of valence electrons in molecules, unpaired electrons in molecules, pairs of valence electrons in atoms, unpaired electrons in atoms. For G3 theory, A=6.386 mhartrees, B=2.977 mhartrees, C=6.219 mhar- trees, D=1.185 mhartrees. Total energy is calculated by adding ZPE from step 2 to this total energy. G3MP2 uses MP2 instead of MP4 to reduce computational cost, G3B3 uses B3LYP to optimize geometry, and G3MP2B3 does both. Neither G2 nor G3 are size consistent. G4 is different from G1-3 in that it depends on cancellation of errors for its high accuracy, rather than on QCI. The mechanism of G4 calculation is the same as that of G3, detailed above. G4 gains accuracy over G3 by: 1. Determining the HF limit for total inclusion of energy 2. Increasing d-polarization to 3d on first row atoms, 4d on second row atoms 3. Replacing QCISD(T) with CCSD(T) in HLC 4. Calculating geometry and ZPE at B3LYP/6-31G(2df,p) level 5. Including two more HLC parameters a. A corrects inaccuracies for pairs of electrons in radical molecules that also have an ion b. E corrects inaccuracies for molecules with a valence 1s pair of electron The absolute energy error of G4 is 0.83 kcal/mol, which is lower than the 1.02 kcal/mol of G3. The error in enthalpies of formation for non-hydrogen species is significantly reduced from G3 as well CBS: Complete Basis Set Extrapolation Energy calculations run with this method are computationally cheap and have high accuracy. It is obvious that the most accurate energy is that of the solution of the Schrödinger Equation- or at an infinitely large basis set. CBS methods approximate an infinitely large basis set by combining energies from many lower-level theories. The most accurate CBS method is CBS-APNO (Atomic Pair Natural Orbital) which has a mean absolute deviation of only 0.5 kcal/mol. This is nearly twice as accurate as all previously

4 described methods. This increased accuracy comes at a computational cost, though. The computational difficulties arise from the expensive QCISD(T)/6-311+G(2df,p) calculation. This method is only applicable to first row compounds. The next most accurate CBS method is CBS-QB3, which is about two times less accurate than CBS-APNO but is also significantly faster. The fastest (and least accurate) CBS method is CBS- 4M. Its advantage is that it can be used on large systems, which cannot be said for most other high accuracy methods. The primary difference between CBS methods and Gaussian methods is that Gaussian increases accuracy by adding in more and more empirical terms trying to correct for known issues with the models being used, while CBS corrects the energy by trying to extrapolate the basis set to the infinite basis set W1: Weizmann-1 These are by far the most accurate methods with a mean error of only 0.3 kcal/mol. The mechanism of calculation is similar to that of CBS where the basis function is extrapolated to infinity. The high accuracy is due to the use of very large basis sets: up to cc-pvqz + 2d1g and cc-pv5z + 2d1f and calculations at the CCSD and CCSD(T) level. The variants are W1U, W1BD and W1 RO. The W1BD uses BD instead of Coupled Clusters (CC) and is extremely accurate as well as computationally expensive. The very high level of accuracy can be obtained without the use of any empirical parameters. This is a main point of difference between W1 and Gn methods. Unfortunately, the very high accuracy comes at a very high computational cost, limiting the application of these methods to small molecules Comparisons Computational Cost: W1> G2, CBS-APNO > G4 > G3, G2MP2 > CBS-QB3 > CBS-4M Accuracy: W1 > CBS-APNO > CBS-QB3, G4 > G3 > G2, G2MP2, CBS-4M 23.5 Gaussian keywords Can not specify basis set keywords with any method. Opt=(MaxCyc=n) can be used to modify cycles in optimization. QCISD=maxcyc, CCSD=maxcyc to modify cycles in energy calculations. ReadIsotopes to specify temperature, pressure, scaling factor. Restart to restart calculations using new parameters.

5 23.6 Examples As an example, consider proton affinity (PA) of water. PA is calculated as the change in energy (at 0K) between the molecule with and without an extra proton. The following input calls for a G2 calculation on water: %chk=h2o.chk #T G2 Test G2 on H2O - calculating proton affinity 0 1 H O 1 OH H 2 OH 1 OHO OH = 1.08 OHO = I didn t call for an extra optimization or restart. Notice that I didn t use any extra basis set keywords. The output summary is: Temperature= Pressure= E(ZPE)= E(Thermal)= E(QCISD(T))= E(Empiric)= DE(Plus)= DE(2DF)= G1(0 K)= G1 Energy= G1 Enthalpy= G1 Free Energy= E(Delta-G2)= E(G2-Empiric)= G2(0 K)= G2 Energy= G2 Enthalpy= G2 Free Energy= DE(MP2)= G2MP2(0 K)= G2MP2 Energy= G2MP2 Enthalpy= G2MP2 Free Energy= Notice that at the G2 level, G1 and G2MP2 energies are calculated along with G2 energy. Input of hydronium ion %chk=hydronium.chk #T HF/6-31G(d) G2 Test G2 on H3O+ - calculating proton affinity 1 1 O H 1 r1 H 1 r1 2 a1 H 1 r1 2 a1 3 d1 r1=0.904 a1= d1=

6 Output: Temperature= Pressure= E(ZPE)= E(Thermal)= E(QCISD(T))= E(Empiric)= DE(Plus)= DE(2DF)= G1(0 K)= G1 Energy= G1 Enthalpy= G1 Free Energy= E(Delta-G2)= E(G2-Empiric)= G2(0 K)= G2 Energy= G2 Enthalpy= G2 Free Energy= DE(MP2)= G2MP2(0 K)= G2MP2 Energy= G2MP2 Enthalpy= G2MP2 Free Energy= Find change in energy (at 0K): E = E(H 3 O + ) E(H 2 O) E(H + ) G = [( ) ( )] = kcal/mol The H + has no electrons and no ZPE therefore no energy at 0K. Experimental value: kcal/mol Deviation from experiment: 1.4 kcal/mol

Performance of Hartree-Fock and Correlated Methods

Chemistry 460 Fall 2017 Dr. Jean M. Standard December 4, 2017 Performance of Hartree-Fock and Correlated Methods Hartree-Fock Methods Hartree-Fock methods generally yield optimized geomtries and molecular