Computational SpectroscopyII. ab initio Methods

from part (g) Advanced Methods for Thermochemistry

Chemistry 713

Updated: February 28, 2008

Advanced Methods for Thermochemistry

1. Thermodynamics from Gaussian data

2. Rate constants from Gaussian data

3. “High accuracy” extrapolation methods- G1, G2, G3, CBS-4M, CBS-Q, CBS-QB3

4. Condensed phase methods- treatments of solvation

Thermodynamic quantities available from a Gaussian FREQ calculation

# freq hf/3-21+g geom=connectivity…Vinyl fluoride frequency… ------------------- - Thermochemistry - ------------------- Temperature 298.150 Kelvin. Pressure 1.00000 Atm.... This molecule is an asymmetric top. Rotational symmetry number 1. Rotational temperatures (Kelvin) 3.05888 .50432 .43294 Rotational constants (GHZ): 63.73686 10.50837 9.02106 Zero-point vibrational energy 124526.4 (Joules/Mol) 29.76253 (Kcal/Mol) Warning -- explicit consideration of 1 degrees of freedom as vibrations may cause significant error Vibrational temperatures: 739.91 1123.38 1431.13 1558.02 1624.71 (Kelvin) 1789.79 2083.59 2245.96 2657.35 4809.86 4928.00 4962.46 Zero-point correction= .047430 (Hartree/Particle) Thermal correction to Energy= .050672 Thermal correction to Enthalpy= .051616 Thermal correction to Gibbs Free Energy= .023092 Sum of electronic and zero-point Energies= -175.909354 Sum of electronic and thermal Energies= -175.906112 Sum of electronic and thermal Enthalpies= -175.905167 Sum of electronic and thermal Free Energies= -175.933691

E (Thermal) CV S KCal/Mol Cal/Mol-Kelvin Cal/Mol-Kelvin Total 31.797 9.146 60.033 Electronic .000 .000 .000 Translational .889 2.981 37.405 Rotational .889 2.981 21.503 Vibrational 30.019 3.185 1.126 Vibration 1 .869 1.218 .623

Q Log10(Q) Ln(Q) Total Bot .238974E-10 -10.621649 -24.457250 Total V=0 .156464E+12 11.194414 25.776090 Vib (Bot) .174410E-21 -21.758428 -50.100631 Vib (Bot) 1 .315520E+00 -.500974 -1.153534 Vib (V=0) .114192E+01 .057635 .132710 Vib (V=0) 1 .109123E+01 .037916 .087305 Electronic .100000E+01 .000000 .000000 Translational .122716E+08 7.088902 16.322799 Rotational .111655E+05 4.047877 9.320581

A portion of the G03 output “.log” file.

(cont’d)….

Thermodynamic quantities

Reactants Transition state (‡) Products

Difference in electronic energy

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ΔU00

Zero point energy

Zero point energy

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ΔU2980

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H = U + PV , ∴ ΔHT0 = ΔUT

0 + Δ PV( ) for changes at temp T.

G = H − TS, ∴ ΔGT0 = ΔHT

0 + TΔST0, Kc = exp −ΔGT

0 RT( )

Avg Thermal Energy

Avg Thermal Energy

See Cramer, p 355 ff.D. A. McQuarrie, Statistical Mechanics

Rates of Chemical Reaction

Reactants Transition state (‡) Products

Difference in electronic

energy

Zero point energy

Zero point energy

Avg Thermal Energy

Average Thermal Energy of molecules that react

E0Barrier to reaction

EaActivationEnergy

Arrhenius rate:

Transition state theory:

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k =kBT

h

Q

Qreactants

exp−E0

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

‡

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k = Aexp−Ea

kBT

⎛

⎝ ⎜

⎞

⎠ ⎟

See Cramer, p 519 ff.J. I. Steinfeld, J.S. Francisco, W. L. Hase, Chemical Kinetics and Reaction Dynamics

For aA + bB products, the partition functions are

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Qreactants ≡ QA( )a

QB( )b

and Q which does NOT include the reaction coordinate.

‡Transition state theory, when carefully applied with good ab initio data, can give very good rate constants. The Arrhenius expression is used mainly as a means of fitting experimental data, and is NOT commonly used to generate calculated rates from ab initio data.

Gaussian-1, Gaussian-2, Gaussian-3, and CBS Theories Basis set and correlation extrapolation methods calibrated on a large number of species

to get the thermochemistry right. G1, G2, G3, and several variants developed by Pople are available in G03.

G1 & G2 in Foresman & Frisch, pp 150-153 G3 in Curtis et al., J. Chem. Phys. 109, 7764 (1998). G2 gets 148 enthalpies with absolute deviations from experiment of 1.56 kcal/mol, and G3

within 0.94 kcal/mol. “Complete Basis Set” (CBS) methods (CBS-4M, CBS-Q, CBS-QB3) developed by

Peterson and collaborators are also available in G03. Foresman & Frisch, pp 154-160. CBS-Q is a little better than G2; CBS-QB3 is comparable to G3, but faster.

The better levels of these “high accuracy methods” are demanding on CPU time. Compromise on the level is necessary for larger systems.

The calculations involved are fully ab initio, but the design of the extrapolation methods is empirical, based on a a range of test systems

Gives some semiempirical character to these “high accuracy methods” Work well for the kinds of systems and for the properties on which they were

calibrated, but otherwise caveat emptor.

Details of the G1 and G2 methods All terms must be added to get the final energy (Foresman & Frisch, p 152). G3 is similar but is more complicated (Curtis et al., J. Chem. Phys. 109, 7764 (1998) ).

G2 add these

G1 add theseenergies

Foresman & Frisch, p 152

G2 & G3 accuracy

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Curtis et al., J. Chem. Phys. 109, 7764 (1998).

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CBS vs G3 CBS-QB3 is comparable in

accuracy to G3, but runs faster. G3(MP2) is almost as good and is

faster than CBS-B3.

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(protonaffinity)

Montgomery, Frisch, Ochterski, and Peterson, J. Chem. Phys. 112, 6532 (2000).

(electronaffinity)

(ionizationpotential)

(heat offormation)

Solvation methods Solvation of the subject molecules can have an important, sometimes dramatic, effect on

calculated properties, including relative energies, molecular geometries and spectra. Different kinds of solvents can have very different effects: hydrogen-bonding solvents vs

polar solvents vs non-polar solvents. Explicit (or “discrete”) methods treat a collection of individual solvent molecules as part

of the molecular system. Treat interactions between solute and solvent molecules explicitly. Configurational sampling is necessary. Computationally demanding: Usually only molecular mechanics methods are feasible for this

approach. Implicit (or “continuum”) methods treat only the core molecular system with ab initio

methods; the solvent is replaced by an average field with a suitable dielectric constant etc.

Create a cavity in the solvent of the right size and shape for the molecular system. Does not add significantly to the cost of the calculation.

Hybrid approaches treat a small number of solvent molecules explicitly and the rest of the solvent implicitly.

See the figure above. (Mark Gordon, www.msg.ameslab.gov/tutorials/solvation.pdf)

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For an extended discussion of solvation models, see Cramer chapter 11 for implicit models and chapter 12 for explicit models.

Solvation Methods in G03 Self-Consistent Reaction Field models are accessed with the SCRF keyword.

SCRF=dipole: Onsager model (simplest) Dipole-induced dipole interaction Supply the radius of the cavity and the dielectric constant of the solvent.

SCRF=PCM: The Polarized Continuum Model treats the cavity as the union of interlocking spheres on each atom numerically integrates the solvent polarization

SCRF=IPCM: The Isodensity PCM model allows for optimizing the cavity shape.

SCRF=SCIPM: The Self-Consistent IPCM model includes the effect of solvation in the electronic SCF solution.

Solvation models require care and some sophistication for proper usage; even then chemical accuracy on solvation effects may be difficult to achieve.

For a detailed description, see the Gaussian 03 User’s References, p 203 and the references listed there.

A case study on solvation effects(2-nitrovinyl)amine: NH2-CH=CH-NO2

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Transition state

Foresman et al., J. Phys. Chem. 100, 16098 (1996).

Syn

Anti

Appendix: The z-matrix%chk=vinylflorifde test.chk%mem=6MW%nproc=1# opt hf/3-21g geom=connectivity

Test 1 vinyl fluoride

0 1 C H 1 B1 H 1 B2 2 A1 C 1 B3 3 A2 2 D1 H 4 B4 1 A3 3 D2 F 4 B5 1 A4 3 D3

B1 1.07000000 B2 1.07000000 B3 1.35520000 B4 1.07000000 B5 1.35000000 A1 120.00000000 A2 120.00000000 A3 120.00000000 A4 120.00000000 D1 180.00000000 D2 0.00000000 D3 -180.00000000

Charge = 0

Spin multiplicity

= 1

Z-matrix

Atoms are numbered 1 to 6 from topto bottom.

The bond length from atom 2 (H) to atom 1 is B1.

The bond angle 3-1-2 is A1.

The dihedral angle 4-1-3-2 is D1.

Here are the values of the variables that appear in the z-matrix. One can also put the numerical data directly into the z-matrix.

The z-matrix appears in the Gaussian input files (.com or .gjf) to specify the initial molecular geometry