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Statistical Mechanics and MultiScale Simulation Methods
ChBE 591-009
Prof. C. Heath Turner
Lecture 04
• Some materials adapted from Prof. Keith E. Gubbins: http://gubbins.ncsu.edu
• Some materials adapted from Prof. David Kofke: http://www.cbe.buffalo.edu/kofke.htm
Introduction to Quantum Chemistry
Basis Sets:
• Split-valence: 3-21G, 6-31G, 6-311G, cc-pVDZ, cc-pVTZ (contraction
coefficients optimized for HF and for e- correlation methods). Allow orbitals to
change size but not shape.
• Polarization Functions:
• Add 1 quantum number of higher angular momentum (than
valence orbitals).
• In practice: adds next level of unoccupied orbitals:
• adds p GTOs to H
• adds d GTOs to first and second row elements
• adds f GTOs to transition metals
• Why? Increases flexibility of the wave function. More important for
molecules than for individual atoms – neighbors and bonds can
polarize the e- in molecules.
• Examples: water, ammonia. Without polarization, HF predicts
planar geometry for NH3.
Introduction to Quantum Chemistry
Polarization of a p-orbital:1
Application to the water molecule:2
1.
2.
A. R. Leach, Molecular Modelling: Principles and Applications, 2nd ed. Prentice Hall (2001).
C. J. Cramer, Essentials of Computational Chemistry, Wiley (2004).
Introduction to Quantum Chemistry
Experimental Bond Angle of H2O (gas phase) = 104.5º
Lets try calculations with and without polarization functions (using Gaussian03)…
Introduction to Quantum Chemistry
Input file #1:
%chk=che651_water_1.chk
%mem=200MB
%nproc=1
# opt hf/6-31g geom=connectivity
Calculation w/o polarization
0 1
O
H
H
1
1
B1
B2
A1
1 2 1.0 3 1.0
2
3
1.0
1.0
100.0
B1
B2
2
A1
Introduction to Quantum Chemistry
Input file #2:
%chk=che651_water_2.chk
%mem=200MB
%nproc=1
# opt hf/6-31g(d,p) geom=connectivity
Calculation w/o polarization
0 1
O
H
H
1
1
B1
B2
A1
1 2 1.0 3 1.0
2
3
1.0
1.0
100.0
B1
B2
2
A1
Introduction to Quantum Chemistry
Experimental Bond Angle of H2O (gas phase) = 104.5º
Lets try calculations with and without polarization functions (using Gaussian03)…
• HF/6-31G: bond angle = 111.5º
• HF/6-31G(d,p): bond angle = 105.9 º
Introduction to Quantum Chemistry
Alternative: “FLOGOs” floating Gaussian orbitals (shown on left):1
1. C. J. Cramer, Essentials of Computational Chemistry, Wiley (2004).
Introduction to Quantum Chemistry
Polarization (nomenclature):
• 6-31G(d) OR 6-31G* = adds d functions to polarize the p functions on
the “heavy” atoms.
• 6-31G(d,p) OR 6-31G** = adds d functions to polarize the p functions
on the “heavy” atoms AND adds p functions to polarize H (and He)
atoms.
• 6-31G(3d2fg,2pd) = heavy atoms polarized by 3 sets of d functions, 2
sets of f functions, and a set of g functions, and H polarized by 2 sets of
p functions, and one set of d functions. (This basis set would be
‘unbalanced’)
Gaussian03 Calculation
%chk=CH3OH.chk
%mem=6MW
%nproc=1
# opt rhf/6-31g geom=connectivity
Title Card Required
0 1
C
H
H
H
H
O
1
2
1
1
1
B1
B2
B3
B4
B5
A1
A2
A3
A4
D1
D2
D3
1.07000000
1.51320851
1.07000000
1.97020303
1.43000000
44.99999991
144.73561047
117.34791957
90.00000001
0.00000000
180.00000000
180.00000000
1 2 1.0 3 1.0 4 1.0 6 1.0
2
3
4
5 6 1.0
6
B1
B2
B3
B4
B5
1
2
4
4
A1
A2
A3
A4
3
3
3
D1
D2
D3
Gaussian03 Results
Methanol Molecule
-114.64
-114.66
Energy
-114.68
-114.70
-114.72
-114.74
-114.76
-114.78
-114.80
-114.82
6-31G
6-31G(d)
6-31G(d,p)
Basis Set
6-311G(d,p)
Gaussian03 Results
Methanol Molecule
1.90
Dipole Moment
1.85
1.80
1.75
1.70
1.65
1.60
1.55
6-31G
6-31G(d)
6-31G(d,p) 6-311G(d,p)
Basis Set
Exp
Gaussian03 Results
Methanol Molecule
110
C-OH Bond Angle
109
108
107
106
105
104
103
102
6-31G
6-31G(d)
6-31G(d,p) 6-311G(d,p)
Basis Set
Exp
Alabama Supercomputer Center
There are PC and Macintosh versions of the SSH Secure Shell Client. If secure shell is not installed
on your machine, contact your system administrator, or look into the PuTTY software, available at
http://www.chiark.greenend.org.uk/~sgtatham/putty/ The PuTTY package allows ssh connections, and
PSCP or PSFTP can be used to transfer files to or from the supercomputers.
Once you are logged-in, here are some useful Unix commands:
• cd ~ (change to home directory)
• pwd (check to see the current directory)
• ls (list the files in the current directory)
• ls –l (list the files in the current directory and show the details)
• cp filename1 filename2 (copy ‘file1’ to ‘file2’)
• cp filename1 dir\filename2 (copy ‘file1’ to ‘file2’ in directory ‘dir’)
• qstat –u UserID (show the jobs running for UserID)
• qdel job# (kill the simulation labeled ‘job#’)
• exit (log out of the machine)
• vi filename (view the file named ‘filename’), ‘:q!’ will quit
• man command (show how to use the command ‘command’), ‘q’ will quit
Gaussian03 Keywords
For a complete list of the G03 keywords, go to the website
http://www.gaussian.com and look under ‘Tech Support’
For single-point energy calculations, here are some useful commands:
• Stable (test for stability of the wavefunction and SCF calculation, tests open shell
and relaxes symmetry constraints)
• Stable=Opt (test for stability of SCF and reoptimize the wavefunction, if needed.
This is a not a geometry optimization.)
• Guess=Mix (mix HOMO and LUMO within the wavefunction. Often useful for
producing a UHF wavefunction for singlet system.)
• UHF (unrestricted Hartree-Fock calculation, open shell)
Introduction to Quantum Chemistry
Diffuse Functions:
• Account for ‘weakly’ bound electrons (excited states, anions, lone
pairs). Normally, e- density is localized near the nucleus in previous
basis sets.
• 6-31+G = one additional set of s and p Gaussian functions on ‘heavy’
atoms.
• 6-31++G = adds one additional diffuse s function on H
• Have an exponent about ¼ smaller than the smallest valence
exponent.
• ‘aug’ prefix in aug-cc-pVnZ indicates addition of diffuse functions (for
each angular momentum already present)
• Necessary for calculating acidities and electron affinities.
• In absence of these specific instances, no strong reason to include
diffuse functions.
Introduction to Quantum Chemistry
The HF Limit:
• Can be extrapolated using the
cc-pVnZ and cc-pCVnZ basis
sets (Dunning's correlation
consistent basis sets):
• Other approximations – additivity assumption:
E[HF/6-311++G(d,p)] ≈ E[HF/6-31G] +
{E[HF/6-31G(d,p)] –
{E[HF/6-311G] –
{E[HF/6-31++G] –
E[HF/6-31G]} +
E[HF/6-31G]} +
E[HF/6-31G]}
Time Savings = (2044 + 3784 + 2944)/5424 = 0.43
Introduction to Quantum Chemistry
Effective Core Potentials:
• Heavy elements are challenging for MO theory (many e-, many basis functions).
• Atom nuclear-electronic ‘core’ is approximated.
• Relativistic effects can be incorporated (important for core electrons in heavy
elements).
• How many e- should be included in ‘core’?
• large-core ECP = include all but valence e- in the core
• small-core ECP = scale back approximation to next lower shell. Usually worth
the added computational cost.
• Most widely used pseudopotentials – Los Alamos National Labs (LANL) ECPs,
Stuttgard/Dresden (SDD)
Introduction to Quantum Chemistry
Property Calculations (HF):
• Energetics:
• Lack of electron correlation – any process that involves the change in the total
# of paired e- is not accurately calculated (heats of formation), isomerizations,
and most other changes in bonding.
• Conformational changes predicted fairly well, such as torsional barriers.
• Cancellation of errors – basis set incompleteness and neglecting e- correlation
introduce errors of opposite sign, sometimes appearing to give accurate results
(in particular, polarized double-zeta basis sets at HF level).
• Protonation/deprotonation energies fairly well estimated.
• Basis Set Superposition Errors (BSSE) – borrowing basis functions within a
complex, resulting in an artificial lowering of the energy.
Introduction to Quantum Chemistry
Property Calculations (HF):
• Geometry:
• With appropriate basis set HF does a good job for minimum-energy structures.
• Bond lengths, bond angles, and dihedral angles are reproduced within a few
percent. Bond length err by underpredicting.
• Polarization functions required in systems with hypervalent bonding,
heteroatoms with single lone pairs.
• Crowding of non-bonded e- can cause e- to become important
• Dative bonds poorly described (both e- in the bonding pair come from only one
of the atoms).
• TS structures not well reproduced.
• Non-bonded complexes not well described – distances unrealistically large.
Hydrogen bonding is usually fairly accurate, due to cancellation of errors.
Introduction to Quantum Chemistry
Property Calculations (HF):
• Charge Distributions:
• Dipole moment (e- + nuclear contribution):
M
nuclear   Z AR A
A1
 N

electronic   d 0    ri 0
 i 1

• Quadrupole moment, octopole moment, etc.
• Scaling factors can be used to correct for small basis sets and e- correlation.
Overestimated by 10-25%, molecules predicted to be too polar.
• Sensitive to optimized geometry.
• Fairly insensitive to basis set size beyond valence-double-zeta.
• Population analysis – assign partial charges to individual atoms (somewhat
arbitrary) – Mulliken population analysis, Löwdin population analysis, natural
population analysis.