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High Accuracy Energy Methods 181 Gaussian Methods he method keywords G1 G2 G2MP2 G3 G3MP2 G3B3 G3MP2B3 G4 and G4MP2 perform high accuracy complex energy computations in Gaussian Essentially this means that ID: 21701

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18. High Accuracy Energy Methods 18.1 Gaussian Methods he 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 correcti on 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). (129) G1 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 (dissoc iation), ionization energies, and electron affinities. This method uses MP2 FU/6 31G*. The inclusion of FU means that all electrons are considered in energy calculations, not just valence electrons. The final energy is calculated by adding a zero point ene rgy (ZPE) that has been calculated using HF/6 3lG* and scaled by the scaling factor 0.8929. 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)

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2. More accurate for hypervalent species 3. More acc urate electron affinities for row 2 atom containing molecules (44% better) 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 141.3 kcal/mol in G1 to 137.5 kcal/mol in G2 a big difference! G2MP2 uses MP2 instead of MP4 in calculations, significantly decreasing com putational 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 potentia l. 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 0.8929 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 ca lculating 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 sec ond row and less on first, and

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core 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 ogether 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. HL C 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 a. For G3 theory, =6.386 mhartrees, =2.977 mhartrees, =6.219 mhar trees, =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 lim it 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. $FRUUHFWVLQDFFX racies 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 nonhydrogen species is significantly reduced from G3 as well. 17.2 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.

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The most acc urate CBS method is CBS APNO (Atomic Pa ir Natural Orbital) which has a mean absolute deviation of only 0.5 kcal/mol. This is nearly twice as accurate as all previously described methods. This increased accuracy comes at a computational cost, though. The co mputational 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 . 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 extr apolate the basis set to the infinite basis set. 17.3 W1 : Weizmann 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 infini ty. 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 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. The very high accuracy comes at a very high computational cost, limiting t he application of these methods to small molecules. 17.4 Comparisons Computational Cost: W1> G2, CBS APNO > G4 > G3 > CBS QB3 CBS Accuracy: W1 > CBS APNO > CBS QB3, G4 > G3 > G2 CBS

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17.5 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. 17.6 Sample Input and Output Example calculation of the proton affinity of water: Input: %chk=H2o.chk #T G2 Test G2 on H2O calculating p roton affinity 0 1 O 1 OH H 2 OH 1 OHO OH = 1.08 OHO = 107.5 ,GLGQWFDOOIRUDQH[WUDRSWLPL]DWLRQRUUHVWDUW1RWLFHWKDW,GLGQWXVHDQ\H[WUDEDVLVVHW keywords. The output summary is: Temperature= 298.150000 Pressure= 1.000000 E(ZPE)= 0.020518 E(Thermal)= 0.023354 E(QCISD(T))= 76.276068 E(Empiric)= 0.024560 DE(Plus)= 0.010833 DE(2DF)= 0.037392 G1( 0 K)= 76.328335 G1 Energy= 76.325499 G1 Enthalpy= 76.324555 G1 Free Energy= 76.345931 E(Delta G2)= 0.008273 E(G2 Empiric)= 0.004560 G2(0 K)= 76.332048 G2 Energy= 76.329212 G2 Enthalpy= 76.328268 G2 Free Energy= 76.349645 DE(MP2)= 0.054454 G2MP2(0 K)= 76.330004 G2MP2 Energy= 76.327169 G2MP2 Enthalpy= 76.326225 G2MP2 Free Energy= 76.347601 Notice that at the G2 level, G1 and G2MP2 energies are calculated along with G2 energy.

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Input of hydronium ion %chk=hydronium.chk #T HF/6 31G(d) G2 Test G2 on H3O+ calculating proton affi nity 1 1 O H 1 r1 H 1 r1 2 a1 H 1 r1 2 a1 3 d1 r1=0.904 a1=114.1859 d1= 133.95 Output: Temperature= 298.150000 Pressure= 1.000000 E(ZPE)= 0.032782 E(Thermal)= 0.035691 E(QCISD(T))= 76.562005 E(Empiric)= 0.024560 DE(Plus)= 0.001461 DE(2DF)= 0.033423 G1(0 K)= 76.588667 G1 Energy= 76.585758 G1 Enthalpy= 76.584814 G1 Free Energy= 76.607772 E(Delta G2)= 0.007838 E(G2 Empiric)= 0.004560 G2(0 K)= 76.591945 G2 Energy= 76.589036 G2 Enthalpy= 76.588092 G2 Fre e Energy= 76.611051 DE(MP2)= 0.040685 G2MP2(0 K)= 76.589908 G2MP2 Energy= 76.586999 G2MP2 Enthalpy= 76.586055 G2MP2 Free Energy= 76.609013 Find change in energy Find change in zero point energy 0.032782 ( 0.020518) =0.012264 Calculate proton affinity (notice the factor 5/2 RT where do es it come from?) Experimental value: 165.1 kcal/mol Deviation from experiment: 1.07 kcal/mol