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Part I: Introduction to Computational Methods Used in Gaussian 09

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

2 Atomic Units Physical quantity Atomic units Values in SI units Length a0 (Bohr) 40ħ2/mee2 =  m Mass me  kg Charge E  C Energy a (Hartree) mee4/(40)2ħ2 =  J Angular momentum ħ h/2 =  Js Permittivity 40  C2/Nm2 The Hamiltonian operator for the hydrogen atom: In atomic units, the Schrödinger equation for this atom is simplified into from

3 Energy Conversion Table
hartree eV cm-1 kcal/mol kJ/mol oK J Hz 1 43.60 x 10-19 x 10+15 x 10-19 x 10+14 x 10-6 x 10-4 x 10-23 x 10+10 6.95 x 10-21 x 10+13 83.593 1.66 x 10-21 x 10+12 x 10-23 x 10+10 2.294 x 10+17 x 10+18 x 10+22 1.44 x 10+20 6.02 x 10+20 x 10+22 x 10+33 x 10-16 x 10-15 x 10-11 x 10-14 x 10-11 x 10-34

4 The Atomic Units Given in Output Files of Gaussian 09
In a unit of Å In a unit of a hartree =  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
ADMP AM1 Amber B3LYP BD BOMD CacheSize CASSCF CBSExtrapolate CCD, CCSD Charge ChkBasis CID, CISD CIS, CIS(D) CNDO Complex Constants Counterpoise CPHF Density DensityFit DFTB Dreiding EOMCCSD EPT ExtendedHuckel External ExtraBasis ExtraDensityBasis Field FMM Force Freq Gen, GenECP GenChk Geom GFInput GFPrint Guess GVB HF Huckel INDO Integral IOp IRC

9 Named Keywords in Gaussian 09
IRCMax LSDA MaxDisk MINDO3 MNDO Name NMR NoDensityFit ONIOM Opt Output OVGF PBC PM3 PM6 Polar Population Pressure Prop Pseudo Punch QCISD Restart Route (#) SAC-CI Scale Scan SCF SCRF SP Sparse Stable Symmetry TD Temperature Test TestMO TrackIO Transformation UFF Units Volume ZIndo

10 Gaussian 09 Keywords: Keyword Topics and Categories
CBS Methods Density Functional (DFT) Methods G1-G4 Methods Frozen Core Options Molecular Mechanics Methods MP & Double Hybrid DFT Methods Semi-Empirical Methods W1 Methods Link 0 Commands Summary Gaussian 09 User Utilities The FormChk Utility Program Development Keywords Obsolete 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  electron Orbital of the  electrons 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
Maximum Number of atoms in Molecule Computed quantities MM 2000 – 1 million Rough geometrical structure Semiempirical 500 – 2000 Geometrical structure (for organic molecules) HF(DFT) 50 – 500 Energy (also for transition metals) MP2 20 – 50 Energy (weak bonding or H-bond) CCSD(T) 10 – 20 Exact energy CASPT2 < 10 Magnetism (involved in several spin multiplicities)

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 more accurate 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

20 2013 Nobel Prize in Chemistry
The Nobel Prize in Chemistry 2013 was awarded jointly to Arieh Warshel Martin Karplus Michael Levitt "for the development of multiscale models for complex chemical systems" Theoretical and computational Chemistry becomes more important to chemists!

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 Keywords in Gaussian 09: R=restricted Closed shell: HF=hf=RHF=rhf Open shell: UHF=uhf, =ROHF=rohf U=unrestricted

23 HF Keywords in Gaussian 09

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
(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: Unperturbation Hamiltonian Perturbation is applied gradually 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 Keywords in Gaussian 09: R=restricted 2-order perturbation correction Closed shell: RMP2 = MP2 = mp2, … Open shell: UMP2 = ump2, … U=unrestricted

30 MP Keywords in Gaussian 09

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 a nondegenerate ground state, the ground-state molecular energy, wave function and all other molecular electronic properties are uniquely determined by the ground-state electron probability density namely, .” Phys. Rev. 136, (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 The last quantity is a relatively Exchange-correlation potential small term, but is not easy to evaluate accurately. The key to accurate KS DFT calculation of molecular properties 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 Among various approximations, gradient-corrected exchange and 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 (the Perdew 1986 correlation functional)

37 Density Functional (DF) Theory (DFT)
Advantages: Nowadays DFT methods are generally 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 B.S. in Chemistry, 1982, Peking University, Beijing, China Prof. in Computational Chemistry, Present, Department of Chemistry, Duke University

39 DFT Keywords in Gaussian 09

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: G** Computer: Pentium (R) Dual-Core E5400/2GB/500GB SATA Calculated NH3 Structure Methods HF MP2 B3LYP Exptl dNH/Ǻ 1.000 1.012 1.016 1.017 HNH/ 108.8 107.9 108.1 107.5 Time/s 5.0 9.0 6.0 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


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