Transcript 幻灯片 1 - xidian.edu.cn

Part I: Introduction to Computational Methods Used
in Gaussian 09
Atomic Units
Physical quantity
Length
Mass
Charge
Atomic units
a0 (Bohr)
me
E
Energy
a (Hartree)
Angular momentum ħ
Permittivity
40
Values in SI units
40ħ2/mee2 = 5.2918  10-11 m
9.1095  10-31 kg
1.6022  10-19 C
mee4/(40)2ħ2 = 4.3598  10-18 J
h/2 = 1.0546  10-34 Js
1.1127  10-10 C2/Nm2
The Hamiltonian operator for the hydrogen atom:
2
1
1 2 1
2
ˆ
ˆ
ˆ
H  T V  
 (SI units)  -  - (Atomic units)
2m
4 0 r
2
r
In atomic units, the Schrödinger equation for this atom is simplified into
2
1 2 1
1
2
(-  - )  E
 )  E
from (2
r
2m
4 0 r
Energy Conversion Table
hartree
hartre
1
e
cm-1
eV
kcal/mol
kJ/mol
oK
J
Hz
27.2107
219
474.63
627.503
2 625.5
315 777.
43.60 x
10-19
6.57966 x
10+15
8 065.73
23.060 9
96.486
9
11 604.9
1.602 10
x 10-19
2.418 04 x
10+14
1.986 30
x 10-23
2.997 93 x
10+10
6.95 x 10-
1.048 54 x
10+13
eV
0.0367502
1
cm-1
4.556 33 x
10-6
1.239 81 x
1
10-4
0.002 859
11
0.011
962 7
1.428 79
kcal/
mol
0.001 593
62
0.043 363
4
1
4.18400
503.228
kJ/mo 0.000 380
l
88
0.010 364
10
83.593
0.239001
1
120.274
0.000 003
166 78
0.000 086
170 5
0.695 028
0.001 987
17
0.008
314 35
1
2.294 x
10+17
6.241 81 x 5.034 45 x 1.44 x
10+18
10+22
10+20
6.02 x
10+20
7.243 54 x
1
10+22
1.509 30 x
10+33
1.519 83 x
10-16
4.135 58 x 3.335 65 x 9.537 02
10-15
10-11
x 10-14
4.799 30 x 6.625 61
10-11
x 10-34
1
oK
J
Hz
349.757
21
1.66 x 1021
1.380 54
x 10-23
2.506 07 x
10+12
2.083 64 x
10+10
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
Computational Methods Used Frequently
Time-independent Schrödinger equation:
Hˆ  (Tˆ  Vˆ )  E , where
Computational Methods Used Frequently
How to solve Hˆ  (Tˆ  Vˆ )  E ?
Computational Chemistry
Based on Newton equations
Molecular mechanics (MM)
(no electronic effects)
Based on Quantum mechanics
Electronic structure methods (QM)
Including
According force fields: UFF, Dreiding, Amber
(Electronic effects)
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, …
Computational Methods Available in Gaussian 09
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
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
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
Computational Methods Available in GaussView
How to Set up Computational Methods in an Input File of Gaussian
Restricted vs. Unrestricted Calculations
Spin-orbital:  is  i (i), or   i  (i)
Orbital of the  electron
 i  i


Closed shell, all pairs of opposite spin
Spin-restricted calculations
Orbital of the  electrons
 i   i
Open shell, unpaired electrons
Spin-unrestricted calculations
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
Computed quantities
of atoms in Molecule
MM
2000 – 1 million
Semiempirical 500 – 2000
Rough geometrical structure
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
More basis functions
Exact solution = Experimental measurements
Part II: The Hartree-Fock (HF) Method
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
●
Q2    2 d 2   |  2 |2  2 d 2
(A continuous charge distribution)
Potential energy between them:
Q1Q2
Q1 | 2 |2
| 2 |2
v12 

d 2 
d 2

4 0 4 0
r12
r12
The potential energy of interaction electron 1 and the other (N-1)
electrons and nuclei is
1 N | i |2
1
V1 (r1 ,1 , 1 )  V12  V13    V1N   
d i 
r1 i 2 r1i
r1
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Hartree-Fock(HF) Method
1 N | i |2
1
V1 (r1 ,1 , 1 )  V12  V13    V1N   
d i 
r1 i 2 r1i
r1
Central-field approximation
V1 (r1 ,1 , 1 ) can be adequately approximated by a function of r only:
2

0
0
 
V (r ,  ,  ) 
1
1
1
1
V1 (r1 ,1 , 1 ) sin 1d1d1
2

0
0
 
sin 1d1d1
=V1 (r1 )
(Average v(r1,1,1) over angles)
One-electron Hartree-Fock(HF) equation:
1 2
ˆ
H1   1  V1 (r1 )
Hˆ 11  11
2
n
Given   cii ,
i 1
the HF equation becomes the Hartree-Fock-Roothannn equation (HFR).
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
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
Møller-Plesset (MP) Perturbation Theory
-e
(x2,y2,z2)
r12
1,y1,z1)
r1
●
-e r2
●(x
The Hamiltonian operator is
●
1 2 1 2 Z Z 1
ˆ
H  - 1 - 2 -  
2
2
r1 r2 r12
Perturbed system
Separate the Hamiltonian into tow parts:
1 2 1 2 Z Z An exactly solvable problem
0
ˆ
（）
1 H  - 1 - 2 - 
2
2
r1 r2
 1 2 Z   1 2 Z  ˆ0 ˆ0
=  - 1 -  +  - 2   =H1 +H 2
Unperturbed system
r1   2
r2 
 2
Namely, the sum of two hydrogen-Hamiltonians, one for each electron.
+2e
Interparticle distances in He
1
'
ˆ
（2）H  Perturbation
, which is interelectronic interaction
r12
Møller-Plesset (MP) Perturbation Theory
Hamiltonian for the perturbed system:
Hˆ  Hˆ 0  Hˆ '=Hˆ 0   Hˆ '
Perturbation is applied gradually
Unperturbation Hamiltonian
Perturbation Hamiltonian




Hˆ n  H 0  Hˆ '  n  H 0   Hˆ '  n En n
 2 n
 2

 n   n  =0 + n

 2 n
+ 2

 =0
E
En  En  =0 + n

 2 En
+ 2

 =0
2
 =0
 2 En
and
k！
 2
2
 =0
k！
2
 =0
+ 
2！
2
 =0
2!
+ 
Kth-order correction to the
wave function and energy
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
Closed shell: RMP2 = MP2 = mp2, …
Open shell: UMP2 = ump2, …
U=unrestricted
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
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  0 ( x, y, z), namely, E0  E0 0  .”
Phys. Rev. 136, 13864 (1964)
Density functional theory (DFT) attempts to
calculate E0 and other ground-state molecular properties
from the ground-state electron density
0 .
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
hˆ KS (1) iKS (1)   iKS iKS (1)

Z
 r2  
1
hˆ KS (1)   12     
dr2   XC (1)
2
r12
 r1
The last quantity v XC 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 EXC
Exchange-correlation potential

 E XC  r 
 XC r  


 r 
Density Functional (DF) Theory (DFT)

Various approximate functionals EXC  r  are used in molecular
DF calculations. The functional E XC is written as the sum of an
exchange-energy functional E X and a correlation-energy functional EC :
EXC  EX  EC
Among various E XC   approximations, gradient-corrected exchange and
correlation energy functionals are the most accurate.
Commonly used E X and EC
PW86 (Perdew and Wang’s 1986 functional)
EX : B88 (Becke’s 1988 functional)
PW91 (Perdew and Wang’s 1991 functional)
EC : Lee-Yang-Parr (LYP) functional
P86 (the Perdew 1986 correlation functional)
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
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
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
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
Methods
dNH/Ǻ
HNH/
Time/s
HF
1.000
108.8
5.0
MP2
1.012
107.9
9.0
B3LYP
1.016
108.1
6.0
Exptl
1.017
107.5
From the viewpoints of computational accuracy and efficiency, the
DFT method (B3LYP) is better than the HF and MP2 methods
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