Part I: Introduction to Computational Methods Used in Gaussian 09

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Part I Introduction to Computational Methods
Used in Gaussian 09
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Atomic Units
Physical quantity Atomic units Values in SI units
Length a0 (Bohr) 4??0h2/mee2 5.2918 ? 10-11 m
Mass me 9.1095 ? 10-31 kg
Charge E 1.6022 ? 10-19 C
Energy ?a (Hartree) mee4/(4??0)2h2 4.3598 ? 10-18 J
Angular momentum h h/2? 1.0546 ? 10-34 Js
Permittivity 4??0 1.1127 ? 10-10 C2/Nm2
The Hamiltonian operator for the hydrogen atom
In atomic units, the Schrödinger equation for
this atom is simplified into
from
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Energy Conversion Table
hartree eV cm-1 kcal/mol kJ/mol oK J Hz
hartree 1 27.2107 219 474.63 627.503 2 625.5 315 777. 43.60 x 10-19 6.57966 x 1015
eV 0.0367502 1 8 065.73 23.060 9 96.486 9 11 604.9 1.602 10 x 10-19 2.418 04 x 1014
cm-1 4.556 33 x 10-6 1.239 81 x 10-4 1 0.002 859 11 0.011 962 7 1.428 79 1.986 30 x 10-23 2.997 93 x 1010
kcal/mol 0.001 593 62 0.043 363 4 349.757 1 4.18400 503.228 6.95 x 10-21 1.048 54 x 1013
kJ/mol 0.000 380 88 0.010 364 10 83.593 0.239001 1 120.274 1.66 x 10-21 2.506 07 x 1012
oK 0.000 003 166 78 0.000 086 170 5 0.695 028 0.001 987 17 0.008 314 35 1 1.380 54 x 10-23 2.083 64 x 1010
J 2.294 x 1017 6.241 81 x 1018 5.034 45 x 1022 1.44 x 1020 6.02 x 1020 7.243 54 x 1022 1 1.509 30 x 1033
Hz 1.519 83 x 10-16 4.135 58 x 10-15 3.335 65 x 10-11 9.537 02 x 10-14 4.799 30 x 10-11 6.625 61 x 10-34 1
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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
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Computational Methods Used Frequently
Time-independent Schrödinger equation
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Computational Methods Used Frequently
Computational Chemistry
Based on Quantum mechanics
Based on Newton equations
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,

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Computational Methods Available in Gaussian 09
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Named Keywords in Gaussian 09
ADMPAM1AmberB3LYPBDBOMDCacheSizeCASSCFCBSExtrapolateCCD, CCSDChargeChkBasisCID, CISDCIS, CIS(D)CNDOComplexConstantsCounterpoiseCPHFDensityDensityFitDFTBDreiding EOMCCSDEPTExtendedHuckelExternalExtraBasisExtraDensityBasisFieldFMMForceFreqGen, GenECPGenChkGeomGFInputGFPrintGuessGVBHFHuckelINDOIntegralIOpIRC
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Named Keywords in Gaussian 09
IRCMaxLSDAMaxDiskMINDO3MNDONameNMRNoDensityFitONIOMOptOutputOVGFPBCPM3PM6PolarPopulationPressurePropPseudoPunch QCISD RestartRoute () SAC-CIScaleScanSCFSCRFSPSparseStableSymmetryTDTemperatureTestTestMOTrackIOTransformationUFFUnitsVolumeZIndo
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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 Utility Program
Development Keywords Obsolete Keywords and
Deprecated
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Computational Methods Available in GaussView
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How to Set up Computational Methods in an Input
File of Gaussian
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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.
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Application Fields for Various Computational
Methods
Method 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 lt 10 Magnetism (involved in several spin multiplicities)
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Reliable Results from Electronic Structure
Calculations
H-F bond energy calculated at different
computational levels
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Computational RD is Growing in Relative
Importance
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Comparison Among Various Computational Methods
More basis functions
Exact solution Experimental measurements
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Part II The Hartree-Fock (HF) Method
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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
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The Nobel Prize in Chemistry 2013 was awarded
jointly to
Arieh Warshel
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"for the development of multiscale models for
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Theoretical and computational Chemistry becomes
more important to chemists!
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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).
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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(s1)h2 for open shell
  molecules

Rrestricted
Keywords in Gaussian 09

• Closed shell HFhfRHFrhf
• Open shell UHFuhf, ROHFrohf

Uunrestricted
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HF Keywords in Gaussian 09
http//www.gaussian.com/g_tech/g_ur/k_hf.htm
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HF Methods Available in GaussView
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How to Set up HF Methods in an Input File of
Gaussian
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Part III The Møller-Plesset (MP) Perturbation
Method
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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
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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
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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
Rrestricted
Keywords in Gaussian 09

• Closed shell RMP2 MP2 mp2,
• Open shell UMP2 ump2,

Uunrestricted
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MP Keywords in Gaussian 09
http//www.gaussian.com/g_tech/g_ur/k_mp.htm
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MP Methods Available in GaussView
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How to Set up MP Methods in an Input File of
Gaussian
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Part IV The Denisty Functional Theory (DFT)
Method
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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, 13864 (1964)
Density functional theory (DFT) attempts to
and other ground-state molecular properties
calculate
from the ground-state electron density
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Density Functional (DF) Theory (DFT)
The molecular (Hohenberg-Kohn, KS) orbitals can
be obtained from Hohenberg-Kohn theorem
KS orbitals
One-electron KS Hamiltonian
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 molecular properties is to
get a good approximation to
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Density Functional (DF) Theory (DFT)
Various approximate functionals
are used in molecular
DF calculations. The functional
is written as the sum of an
and a correlation-energy functional
exchange-energy functional
approximations, gradient-corrected exchange and
Among various
correlation energy functionals are the most
accurate.
Commonly used
and
PW86 (Perdew and Wangs 1986 functional)
B88 (Beckes 1988 functional)
PW91 (Perdew and Wangs 1991 functional)
Lee-Yang-Parr (LYP) functional
P86 (the Perdew 1986 correlation functional)
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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,

Rrestricted
Uunrestricted
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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
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DFT Keywords in Gaussian 09
http//www.gaussian.com/g_tech/g_ur/k_dft.htm
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DFT Methods Available in GaussView
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How to Set up DFT Methods in an Input File of
Gaussian
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Dependence of Computational Accuracy and Time on
Computational Methods
Computational conditions Basis sets
6-31G Computer Pentium (R) Dual-Core
E5400/2GB/500GB SATA
Calculated NH3 Structure
Methods HF MP2 B3LYP Exptl
dNH/? 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
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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