Part I: Introduction to Computational Methods Used in Gaussian 09

Atomic Units

Physical quantity Atomic units

Values in SI units

Length a0 (Bohr) 40ħ2/mee2 = 5.2918 10-11 m

Mass me 9.1095 10-31 kg

Charge E 1.6022 10-19 C

Energy a (Hartree) mee4/(40)2ħ2 = 4.3598 10-18 J

Angular momentum

ħ h/2 = 1.0546 10-34 Js

Permittivity 40 1.1127 10-10 C2/Nm2The Hamiltonian operator for the hydrogen atom: 2

2 2

0

1 1 1ˆ ˆ ˆ - (SI units) - - (Atomic units)2 4 2

H T Vm r r

In atomic units, the Schrödinger equation for this atom is simplified into

21 1(- - )

2E

r

22

0

1(- - )

2m 4E

r

from

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 10+15

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 10+14

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 10+10

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 10+13

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 10+12

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 10+10

J 2.294 x 10+17

6.241 81 x 10+18

5.034 45 x 10+22

1.44 x 10+20

6.02 x 10+20

7.243 54 x 10+22

1 1.509 30 x 10+33

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

Energy Conversion Table

In a unit of Å

In a unit of a

0.00001 hartree = 0.00001 2625.5 kJ/mol = 0.03 kJ/mol

The Atomic Units Given in Output Files of Gaussian 09

Computational Methods Used Frequently

ˆ ˆ ˆ( ) , whereH T V E

Time-independent Schrödinger equation:

Computational Methods Used Frequently

ˆ ˆ ˆHow to solve ( )H T V E ?

Computational Chemistry

Molecular mechanics (MM) Electronic structure methods (QM)

Based on Quantum mechanicsBased on Newton equations

(no electronic effects) (Electronic effects)

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, …

IncludingAccording force fields: UFF, Dreiding, Amber

Including

Computational Methods Available in Gaussian 09

ADMPAM1

AmberB3LYP

BDBOMD

CacheSizeCASSCF

CBSExtrapolateCCD, CCSD

ChargeChkBasis

CID, CISDCIS, CIS(D)

CNDOComplexConstants

CounterpoiseCPHF

DensityDensityFit

DFTBDreiding

EOMCCSDEPT

ExtendedHuckelExternal

ExtraBasisExtraDensityBasisFieldFMMForceFreq

Gen, GenECPGenChkGeom

GFInputGFPrintGuessGVBHF

HuckelINDO

IntegralIOpIRC

Named Keywords in Gaussian 09

IRCMaxLSDA

MaxDiskMINDO3MNDONameNMR

NoDensityFitONIOM

OptOutputOVGFPBCPM3PM6Polar

PopulationPressure

PropPseudoPunch QCISD

RestartRoute (#)SAC-CI

ScaleScanSCF

SCRFSP

SparseStable

SymmetryTD

TemperatureTest

TestMOTrackIO

TransformationUFFUnits

VolumeZIndo

Named Keywords in Gaussian 09

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 Methods

Link 0 Commands SummaryGaussian 09 User Utilities The FormChk Utility Program Development Keywords Obsolete Keywords and Deprecated

Computational Methods Available in GaussView

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

Restricted vs. Unrestricted Calculations

Spin-restricted calculations Spin-unrestricted calculations

i i

is ( ), or ( )i ii i Spin-orbital:

Orbital of the electron Orbital of the electrons

i i

Closed shell, all pairs of opposite spin Open shell, unpaired electrons

Closed and open shell calculations use an initial R and U, respectively: RHF vs. UHF, RMP2 vs. UMP2, and so on.

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 < 10 Magnetism (involved in several spin multiplicities)

Reliable Results from Electronic Structure Calculations

H-F bond energy calculated at different computational levels

Computational R&D is Growing in Relative Importance

Comparison Among Various Computational Methods

Exact solution = Experimental measurements

More basis functions

Part II: The Hartree-Fock (HF) Method

The Hartree-Fock (HF) approximation constitutes the first step towards more accurate approximations

For point charges and then electrons:

Hartree-Fock (HF) Method

Q1 Q2● ●

22 2 2 2 2 2| |Q d d

(A continuous charge distribution)

2 21 2 1 2 2

12 2 20 0 12 12

| | | |

4 4

Q Q Qv d d

r r

Potential energy between them:

The potential energy of interaction electron 1 and the other (N-1) electrons and nuclei is

2

1 1 1 1 12 13 121 1 1

| |1 1( , , )

Ni

N ii i

V r V V V dr r r

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The Nobel Prize in Chemistry 2013 was awarded jointly to

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Theoretical and computational Chemistry becomes more important to chemists!

Hartree-Fock(HF) Method

Central-field approximation

2

1 1 1 1 12 13 121 1 1

| |1 1( , , )

Ni

N ii i

V r V V V dr r r

can be adequately approximated by a function of r only:1 1 1 1( , , )V r 2

1 1 1 1 1 1 10 01 1 1 1 1 12

1 1 10 0

( , , ) sin( , , ) = ( )

sin

V r d dV r V r

d d

(Average v(r1,1,1) over angles)

One-electron Hartree-Fock(HF) equation:

21 1 1 1

1ˆ ( )2

H V r 1 1 1 1H

Given1

,n

i ii

c

the HF equation becomes the Hartree-Fock-Roothannn equation (HFR).

U=unrestricted

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:

Closed shell: HF=hf=RHF=rhf

Open shell: UHF=uhf, =ROHF=rohf

R=restricted

HF Keywords in Gaussian 09

http://www.gaussian.com/g_tech/g_ur/k_hf.htm

HF Methods Available in GaussView

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

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

2 21 2

1 2 12

1 1 1ˆ - - -2 2

Z ZH

r r r

Interparticle distances in He

Namely, the sum of two hydrogen-Hamiltonians, one for each electron.

The Hamiltonian operator is

0 2 21 2

1 2

1 1ˆ1 - - -2 2

Z ZH

r r （）

'

12

1ˆ2 ,Hr

（ ）

●

●

●

+2e

-e

-er12

r1

r2

(x1,y1,z1)(x2,y2,z2)

Møller-Plesset (MP) Perturbation Theory

Separate the Hamiltonian into tow parts:

2 2 0 01 2 1 2

1 2

1 1 ˆ ˆ= - - + - = +2 2

Z ZH H

r r

which is interelectronic interaction

An exactly solvable problem

Perturbed system

Unperturbed system

Perturbation

0 0ˆ ˆ ˆ ˆ ˆ'= 'H H H H H

Unperturbation Hamiltonian

Hamiltonian for the perturbed system:

Perturbation Hamiltonian

Møller-Plesset (MP) Perturbation Theory

0 0ˆ ˆ ˆ' 'n n n n nH H H H H E

2 2

2=0=0 =0

+ + +2

n nn n

！2 2

2=0=0 =0

+ + +2!

n nn n

E EE E

Perturbation is applied gradually

2 2

2

=0

n

k

！

2 2

2

=0

nE

k

！

andKth-order correction to the wave function and energy

Advantages:

Weakness:

Keywords in Gaussian 09:

Closed shell: RMP2 = MP2 = mp2, …Open shell: UMP2 = ump2, …

R=restricted

U=unrestricted

Møller-Plesset (MP) Perturbation Theory

Locate quite accurate equilibrium geometriesMuch faster than CI (Configuration interaction ) methods

Do not work well at geometries far from equilibriumSpin contamination for open-shell molecules

2-order perturbation correction

MP Keywords in Gaussian 09

http://www.gaussian.com/g_tech/g_ur/k_mp.htm

MP Methods Available in GaussView

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

Part IV: The Denisty Functional Theory (DFT) Method

properties are uniquely determined by the ground-state electron

),,,(0 zyx 000 EE

In 1964, Hohenberg and Kohn proved that

molecular energy, wave function and all other molecular electronic

probability density namely,

Phys. Rev. 136, 13864 (1964)

.”

0E

.0

Density functional theory (DFT) attempts to

and other ground-state molecular properties

from the ground-state electron density

“For molecules with a nondegenerate ground state, the ground-state

calculate

Density Functional (DF) Theory (DFT)

KS orbitals

The molecular (Hohenberg-Kohn, KS) orbitals can be obtained from Hohenberg-Kohn theorem:

)1(

2

1)1(ˆ

)1()1()1(ˆ

212

2

1

21 XC

KS

KSi

KSi

KSi

KS

rdr

r

r

Zh

h

Exchange-correlation potential

r

rEr XC

XC

Density Functional (DF) Theory (DFT)

One-electron KS Hamiltonian Orbital energy

XCv

get a good approximation to XCE

The last quantity

calculation of molecular properties is to

is a relatively

small term, but is not easy to evaluate accurately. The key to accurate KS DFT

Various approximate functionals rEXC

DF calculations. The functional XCE

and a correlation-energy functional

CXXC EEE

Among various XCE

Commonly used XE and CE

:XEPW91 (Perdew and Wang’s 1991 functional)

:CE Lee-Yang-Parr (LYP) functional

are used in molecular

approximations, gradient-corrected exchange and

correlation energy functionals are the most accurate.

PW86 (Perdew and Wang’s 1986 functional)B88 (Becke’s 1988 functional)

P86 (the Perdew 1986 correlation functional)

is written as the sum of an

XEexchange-energy functional :CE

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

Density Functional (DF) Theory (DFT)

Weakness: Fails for very weak interactions (e.g., van der Waals molecules)

Keywords in Gaussian 09:

Closed shell: RB3LYP = rb3lyp, B3PW91 = b3pw91, …

Open shell: UB3LYP = urb3ly, UB3PW91 = ub3pw91, …

Exchange functional

Correlation functional

R=restricted U=unrestricted

Density Functional (DF) Theory (DFT)

B3LYP

Y is abbreviated for Dr.Yang Weitao

B.S. in Chemistry, 1982, Peking University, Beijing, ChinaProf. in Computational Chemistry, Present, Department of Chemistry, Duke University

DFT Keywords in Gaussian 09

http://www.gaussian.com/g_tech/g_ur/k_dft.htm

DFT Methods Available in GaussView

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

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

Dependence of Computational Accuracy and Time on Computational Methods

From the viewpoints of computational accuracy and efficiency, the DFT method (B3LYP) is better than the HF and MP2 methods

Computational conditionsBasis sets: 6-31++G**Computer: Pentium (R) Dual-Core E5400/2GB/500GB SATA

Calculated NH3 Structure

– 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

List of Computational Methods Used in Gaussian