Part I: Introduction to Computational Methods Used in Gaussian 09

1. Part I: Introduction to Computational Methods Used in Gaussian 09

2. Atomic Units The Hamiltonian operator for the hydrogen atom: In atomic units, the Schrödinger equation for this atom is simplified into from

3. Energy Conversion Table

4. The Atomic Units Given in Output Files of Gaussian 09 In a unit of Å In a unit of a 0.00001 hartree = 0.00001  2625.5 kJ/mol = 0.03 kJ/mol

5. Computational Methods Used Frequently Time-independent Schrödinger equation:

6. Computational Methods Used Frequently Computational Chemistry Based on Newton equations Based on Quantum mechanics Molecular mechanics (MM) Electronic structure methods (QM) (no electronic effects) (Electronic effects) Including • According force fields: UFF, Dreiding, Amber Including • Semiempirical methods: Hückel, AM1, PM3, INDO, … • Ab initio methods: HF, post-HF (MP2, CI, CCSD, CASPT2, …) • Density function theory: DFT(B3LYP, …) • Combination of Quantum mechanics and molecular mechanics: QM/MM, …

7. Computational Methods Available in Gaussian 09

8. Named Keywords in Gaussian 09

9. Named Keywords in Gaussian 09

10. Gaussian 09 Keywords: Keyword Topics and Categories CBS Methods Density Functional (DFT) MethodsG1-G4 Methods Frozen Core OptionsMolecular Mechanics Methods MP & Double Hybrid DFT MethodsSemi-Empirical Methods W1 MethodsLink 0 Commands SummaryGaussian 09 User Utilities The FormChk UtilityProgram Development KeywordsObsolete Keywords and Deprecated

11. Computational Methods Available in GaussView

12. How to Set up Computational Methods in an Input File of Gaussian

13. Restricted vs. Unrestricted Calculations Spin-orbital: Orbital of the  electrons Orbital of the  electron Open shell, unpaired electrons Closed shell, all pairs of opposite spin Spin-unrestricted calculations Spin-restricted calculations Closed and open shell calculations use an initial R and U, respectively: RHF vs. UHF, RMP2 vs. UMP2, and so on.

14. Application Fields for Various Computational Methods

15. Reliable Results from Electronic Structure Calculations H-F bond energy calculated at different computational levels

16. Computational R&D is Growing in Relative Importance

17. Comparison Among Various Computational Methods More basis functions Exact solution = Experimental measurements

18. Part II: The Hartree-Fock (HF) Method

19. Hartree-Fock (HF) Method The Hartree-Fock (HF) approximation constitutes the first step towards moreaccurate approximations For point charges and then electrons: Q1 Q2 ● ● (A continuous charge distribution) Potential energy between them: The potential energy of interaction electron 1 and the other (N-1) electrons and nuclei is

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21. Hartree-Fock(HF) Method Central-field approximation can be adequately approximated by a function of r only: (Average v(r1,1,1) over angles) One-electron Hartree-Fock(HF) equation: Given the HF equation becomes the Hartree-Fock-Roothannn equation (HFR).

22. Hartree-Fock(HF) Method Advantages: • Initial, first level predication of the structures and vibrational frequencies for various molecules Weakness: • Poor modeling of the energetics of reactions • Spin contamination [s(s+1)ħ2] for open shell molecules R=restricted Keywords in Gaussian 09: • Closed shell: HF=hf=RHF=rhf • Open shell: UHF=uhf, =ROHF=rohf U=unrestricted

23. HF Keywords in Gaussian 09 http://www.gaussian.com/g_tech/g_ur/k_hf.htm

24. HF Methods Available in GaussView

25. How to Set up HF Methods in an Input File of Gaussian

26. Part III: The Møller-Plesset (MP) Perturbation Method

27. Møller-Plesset (MP) Perturbation Theory -e r12 ● (x1,y1,z1) (x2,y2,z2) r1 ● The Hamiltonian operator is -e r2 ● +2e Interparticle distances in He Perturbed system Separate the Hamiltonian into tow parts: An exactly solvable problem Unperturbed system Namely, the sum of two hydrogen-Hamiltonians, one for each electron. which is interelectronic interaction Perturbation

28. Møller-Plesset (MP) Perturbation Theory Hamiltonian for the perturbed system: Perturbation is applied gradually Unperturbation Hamiltonian Perturbation Hamiltonian Kth-order correction to the wave function and energy and

29. Møller-Plesset (MP) Perturbation Theory Advantages: • Locate quite accurate equilibrium geometries • Much faster than CI (Configuration interaction ) methods Weakness: • Do not work well at geometries far from equilibrium • Spin contamination for open-shell molecules 2-order perturbation correction R=restricted Keywords in Gaussian 09: • Closed shell: RMP2 = MP2 = mp2, … • Open shell: UMP2 = ump2, … U=unrestricted

30. MP Keywords in Gaussian 09 http://www.gaussian.com/g_tech/g_ur/k_mp.htm

31. MP Methods Available in GaussView

32. How to Set up MP Methods in an Input File of Gaussian

33. Part IV: The Denisty Functional Theory (DFT) Method

34. Density Functional (DF) Theory (DFT) In 1964, Hohenberg and Kohn proved that “For molecules with anondegenerateground state, the ground-state molecular energy, wave functionand all other molecular electronic properties are uniquely determined by the ground-state electron probability density namely, .” Phys. Rev. 136, 13864 (1964) Density functional theory (DFT) attempts to calculate and other ground-state molecular properties from the ground-state electron density

35. Density Functional (DF) Theory (DFT) The molecular (Hohenberg-Kohn, KS) orbitals can be obtained from Hohenberg-Kohn theorem: One-electron KS Hamiltonian KS orbitals Orbital energy Exchange-correlation potential The last quantity is a relatively small term, but is not easy to evaluate accurately.The key to accurate KS DFT calculation of molecularproperties is to get a good approximation to

36. Density Functional (DF) Theory (DFT) Various approximate functionals are used in molecular DF calculations. The functional is written as the sum of an exchange-energy functional and a correlation-energy functional approximations, gradient-corrected exchangeand Among various correlation energy functionals are the most accurate. Commonly used and PW86 (Perdew and Wang’s 1986 functional) B88 (Becke’s 1988 functional) PW91 (Perdew and Wang’s 1991 functional) Lee-Yang-Parr (LYP) functional P86 (thePerdew 1986 correlation functional)

37. Density Functional (DF) Theory (DFT) Advantages: Nowadays DFT methods aregenerally believed to be better than the HF method, and in most cases they are even better than MP2 Weakness: Fails for very weak interactions (e.g., van der Waals molecules) Exchange functional Keywords in Gaussian 09: Correlation functional • Closed shell: RB3LYP = rb3lyp, B3PW91 = b3pw91, … • Open shell: UB3LYP = urb3ly, UB3PW91 = ub3pw91, … R=restricted U=unrestricted

38. Density Functional (DF) Theory (DFT) B3LYP Y is abbreviated for Dr.Yang Weitao

39. DFT Keywords in Gaussian 09 http://www.gaussian.com/g_tech/g_ur/k_dft.htm

40. DFT Methods Available in GaussView

41. How to Set up DFT Methods in an Input File of Gaussian

42. Dependence of Computational Accuracy and Time on Computational Methods Computational conditions Basis sets: 6-31++G** Computer: Pentium (R) Dual-Core E5400/2GB/500GB SATA Calculated NH3 Structure From the viewpoints of computational accuracy and efficiency, the DFT method (B3LYP) is better than the HF and MP2 methods

43. List of Computational Methods Used in Gaussian MM: AMBER, Dreiding, UFF force field Semiempirical: CNDO, INDO, MINDO/3, MNDO, AM1, PM3 HF: closed-shell, restricted/unrestricted open-shell DFT: many local/nonlocal functionals to choose MP: 2nd-5th order; direct and semi-direct methods CI: single and double CC: single, double, triples contribution High accuracy methods: G1, G2, CBS, etc. MCSCF: including CASSCF GVB