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
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Martin Karplus
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"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