Practical Molecular Quantum ChemistryGaussian and all that

Herbert Frchtl 14 November 2007

Overview Programs Methods Geometry creation Output analysis Geometry Optimisation, Transition States and Reaction Coordinates Basis sets and ECPs Solvation Properties Density Fitting and Local Correlation Problems and Pitfalls

An Incomplete List of Programs Gaussian NWChem MOLPRO GAMESS-UK GAMESS-US ORCA Dalton Turbomole Jaguar Q-Chem ADF MOPAC (various versions) ArgusLab Hyperchem Spartan deMon MPQC DMol3 Aces-II COLUMBUS Various periodic HF/DFT programs can be used for molecules as well

Gaussian Input Syntax

Checkpoint file (for restart or orbital analysis) Memory requirement (too much may slow calculation down) Number of processors (4 on thin cluster node) Keywords Title (must be present, but has no effect) Charge and multiplicity Geometry (Cartesian or Z-matrix) Empty lines (required) There could be additional information for some methods

Multiple Calculations with Gaussian --link1-- starts new input Need checkpoint file to use previous geometry or wavefunction Need to specify explicitly what to use from previous calculation

NWChem Input Format Job name determines names for temporary files end blocks change settings for future calculations task directive triggers calculation input is handled strictly top-tobottom (only settings above a task have an impact) Python procedures for more complicated structures

MOLPRO Input Format Title required Blocks of input grouped via {} Need to specify all parts of calculation (integrals, HF, ) Powerful but complicated command language (loops, arrays, goto, )

GAMESS-UK Input Top-to-bottom work through input Keywords for setting parameters Execution when ENTER is encountered Compound calculations possible with multiple ENTER directives

Modelling Methods for Molecules

FCI CCSD(T)

Accuracy

MP2

DFT QM/MM

Hartree-Fock

Semiempirical Methods Molecular Mechanics

1

10

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10000

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1000000

Number of Atoms

Molecular Mechanics Parametrised forces between atoms or groups No description of electronic structure Cheap large systems and/or long dynamics simulations possible Not good for change in bond structure Often problems associating atom types (need more than just element) Force fields optimised for certain class of molecule Hear more about it next week! Force Fields: UFF AMBER CHARMM OPLS Dreiding

For dynamics use an MD program (AMBER, DL_POLY, )

Semiempirical Methods Electronic structure with severe approximations and parametrised integrals Originally optimised for small(ish) organic molecules Only PM6 (available in MOPAC 2007) parametrised for all elements ZINDO reasonably good for spectra, not for geometries

Methods: AM1 PM3 PM6 ZINDO Often good for initial optimisation. For many purposes a good Force Field is better. Consider QM/MM

Density Functional Theory All ground state properties can be determined as a functional of the electron density (HohenbergKohn Theorem). This functional is not known. Many model functionals in use. Current functionals do not describe static correlation (London dispersion). Better than Hartree-Fock at similar cost.

Density FunctionalsVWN5 BLYP HCTH BP86Increasing quality and computational cost

LDA local density GGA gradient corrected Meta-GGA kinetic energy density included Hybrid exact HF exchange component Hybrid-meta-GGA

Better scaling with system size Allow density fitting for even better scaling

TPSS M06-L B3LYP B97/2 MPW1K

Meta-GGA is bleeding edge and therefore largely untested (but better in theory) Hybrid makes bigger difference in cost and accuracy Look at literature if somebody has compared functionals for systems similar to yours!

MPWB1K M06

Post-HF Methods MP2

Similar to DFT in total accuracy Describes all kinds of correlation energy Scales n5 with basis functions Best feasible black-box method for small molecules Scaling n7

Many more Methods: MP3, MP4, QCISD, BD, CCSD, CCSDT(Q)

CCSD(T)

MP2 may be necessary in case of dispersion dominated interactions (e.g. -)

QM/MMTreat interesting region with higher accuracy Anything not part of the reaction treated on lower level (typically MM) Boundary atoms usually saturated with H atoms Careful where you cut! Gaussian keyword: ONIOM(HF/6-31G(d):UFF) NWChem: Combine QM and MM modules GAMESS-UK: Interface to CHARMM ChemShell Driver for various QM and MM programs from Daresbury Laboratory

A Typical Calculation 1. Get initial geometry 2. Edit input 3. Run calculation 4. Analyse results

Initial Geometry Creation Convert known geometry from different formatbabel ipdb -og03 .gjfinput format output format

babel H

shows available formats Notation: replace with actual argument

Build molecule with graphical builder

Molden Maestro, GaussView

Z-Matrix editor: if you know your internal coordinates Graphical user interfaces, but can export geometries in different formats Runs on Windows MM / Semiempirical program with GUI Can export different formats (G03 is under calculation, not export)

ArgusLab

More in lecture on visualisation

Analysis of Results Most programs produce text output, containing geometry, energy, properties, error messages, More information in restart file (Gaussian checkpoint file, NWChem runtime database, GAMESS-UK punch file) Local programs/scripts to extract information (different in Edinburgh and St Andrews; ask your administrator) babel can extract geometries (most viewers can display xyz format) grep

NWChem: grep @ shows progress of geometry optimisation Gaussian: search for SCF Done (HF or DFT), Maximum Displacement for geometry optimisation

You still need to check if calculation finished successfully! Visualisation of results: see lecture on visualisation

Geometry Optimisation

Geometry Optimisation Algorithms Steepest Descent

Follow direction of gradient until bottom Calculate new gradient SD + component of previous gradient Avoids zigzagging inherent in SD Calculate gradient + Hessian (2nd derivative) at every point Determine minimum on quadratic surface Gaussian: keyword CalcAll GAMESS keyword: RECALCULATE Expensive! Use for difficult cases

Step size does not follow from algorithm No matrix algebra: can be used for very large systems Often the default for MM methods

Conjugate Gradient

Newton

Geometry Optimisation Algorithms (2) Updated Hessian (Quasi-Newton) Calculate only gradient Update approximate Hessian Various algorithms: DFP, BFGS, EF, possibly helped by geometry DIIS Good initial Hessian important! Defaults: (scaled) unit matrix, Force Field, semiempirical Newton and Quasi-Newton methods MAY converge to saddle point. Check force constants!

Consider calculating Hessian once (possibly on lower level of theory) Gaussian: Opt=CalcFC, CalcHFFC, ReadFC Other programs: calculate frequency

No method can guarantee to find the right minimum

Transition State Optimisation Transition state: 1st order saddle point on potential energy surface Method: follow Eigenvector of negative Eigenvalue uphill all other directions downhillConsiderably more difficult than minimisation Need good Hessian Need starting point where Hessian has correct structure: one negative Eigenvalue

Reaction Coordinate Vector of coordinates along a reaction Most common algorithm: Intrinsic Reaction Coordinate (IRC): steepest descent (in mass-weighted coordinates) from TS Can be done directly in Gaussian, via MOPAC interface in GAMESS-UK Can often be approximated by relaxed PES scan of one internal coordinate

Partial Geometry Optimisation Freeze one or more internal coordinates or the Cartesian coordinates of one or more atoms Uses: starting point for TS search part of relaxed PES scan Keywords: Gaussian: variables, constants (Z-matrix input), ModRedundant (more general) MOLPRO: active, inactive NWChem: variables, constants (Z-matrix input), zcoord (independent of input coordinate system) set geometry:actlist (freeze atoms) GAMESS-UK: variables, constants (Z-matrix input), noopt (freeze atoms)

Coordinate Systems Cartesian Redundant Internal x,y,z coordinates of over-complete set of atoms internal coordinates easy to extract from the created by program, output of any program but can be manipulated difficult to create from good for geometry scratch optimisation A Internal (Z-Matrix) D distances, angles, dihedral angles chemically intuitive C B not unique Dihedral: angle between possibly problematic planes A-B-C and B-C-D for large rings Careful with units:ngstrom, Bohr, picometre?

Simple Z-Matrix Geometry Input (Gaussian)Element symbol Number of first neighbour Distance from 1st neighbour Second neighbour Angle atom neighbour 1 neighbour 2 Third neighbour Dihedral angle atom n1 n2 n3 First 3 atoms need less references

Z-Matrix Geometry Input with Variables (NWChem) Same geometry as before, but dihedral H-C-O-H frozen

Gaussian Basis SetsPople split valence 6-31g 6-31+g* 6-31++g** Pople valence triple zeta 6-311g 6-311+g* 6-311++g** Dunning correlation consistent cc-pvdz cc-pvtz cc-pvqz aug-cc-pvdz aug-cc-pvtz aug-cc-pvqz

Diffuse functions long-distance interactions anions Many more basis sets available Post-HF methods need larger basis than DFT

Polarisation functions Flexibility in angular charge distribution

Angular Dependence of Basis Functions Cartesian ( x, y , z ) = x y z ei j k ( x 2 + y 2 + z 2 )

(m = i + j + k )

3p, 6d, 10f, (overcomplete) Spherical (r , , ) = Ylm e ( r 2 )

3p, 5d, 7f, Use the set from original publication of basis set Not all programs allow both sets Generally: more modern basis sets use spherical harmonics

Basis Set Superposition Error (BSSE) Dimerisation energy: E = EAB (EA+EB) but: In dimer, atom A can use basis functions on atom B and vice versa Counterpoise Procedure: calculate energy of monomer A with basis of A + ghost atom B More complicated if A, B are molecules/fragments

A

B

A

B

A

B

Other Basis Set Types For ab initio (Hartree-Fock and post-HF) methods on reasonably large systems, Gaussian basis sets are the only viable option. DFT gives a bit more choice: Slater functions Purely numerical basis For periodic systems, but Plane waves applicable to molecules. See lecture by Carole Plane waves + spherical (PAW, FLAPW, GPW, ) Morrison Generally only one type in one program (except NWChem: Gaussian, plane wave, PAW)

Electron Core Potentials (ECPs), Pseudopotentials Core orbitals are difficult to describe: Cusp at nucleus Relativistic effects in heavy atoms They are largely unaffected by (and dont affect) bonding Replace core orbitals with potential that looks like a core to valence electronsParticularly important for plane wave basis sets See separate lecture

ECPs in Practice Gaussian: specify basis set that contains ECP Most other programs: specify ECP and basis separately (may have same name) All programs allow explicit ECP input

GAMESS-UK

NWChem MOLPRO

Solvation Explicit Solvent Continuum Solvation Models+ + +

-- -

Expensive Solvent may be treated at lower level (QM/MM)

Molecule in cavity inside polarisable medium with given dielectric constant Different methods available PCM COSMO

Solvent treatment is essential for Zwitterions and many ions Geometry optimisation more difficult with both approaches

Infrared SpectroscopyFrequencies are Eigenvalues of mass-weighted force constant matrix (Hessian) Harmonic approximation (and therefore too high) Anharmonic frequencies possible, but expensive (Gaussian keyword Frequency=Anharmonic)

UV/Visible SpectroscopyElectronically excited states vertical excitation energies HOMO-LUMO gap (Koopmans Theorem) Bad; virtual HF/DFT orbital energies unreliable, no orbital relaxation ZINDO semiempirical, limited selection of atoms fast, qualitatively OK CIS HF based; rather inaccurate (but better than Koopman) TD-DFT DFT equivalent of CIS (but founded in different theory); better than CIS CIS(D) CIS with approximate doubles based on MP2; accurate but expensive

NMR PropertiesShielding Tensor chemical shift Spin-Spin couplings Expensive! Requires good basis set at nucleus Consider uncontracting basis functions or (in Gaussian) NMR=(SpinSpin,Mixed) (uncontracts basis and adds functions around nucleus for part of the calculation) Not available in MOLPRO and GAMESS-UKAtom Isotropic shielding in ppm Needs to be compared to value from reference compound optimised and calculated with same method and basis set to get chemical shift. For C and H: TMS as reference.

Thermochemistry Gaussian frequency calculation gives zero-point correction to Energy, Enthalpy and Gibbs Free Energy Properties calculated at 298.15K (default) or userspecified temperature MOLPRO keyword: THERMO,sym= Gaussian can determine thermodynamic reaction properties in a model chemistry (CBS-QB3, G2, ) (expensive!)

Atomic Charges There is no such thing as an atomMulliken Projection of electron density on AO basis Calculated by default by most programs Not very reliable Diffuse basis functions make things worse! NBO Natural Bond Orbitals Built-in in Gaussian GAMESS-UK has direct interface to NBO program NWChem produces input for NBO program AIM Atoms in Molecules Cut molecule at surface of minimum flux Requires separate program to calculate (AIMPAC)

Density Fitting / RI Methods 4-centre 2-electron integral

formal n4 scaling of HF, DFT; worse for post-HF methods Resolution of the Identity (RI)

1 1 ij kl = i (r1 ) j (r2 ) k (r1 )l (r2 )dr1dr2 r r1 r2

Small error due to incompleteness of fitting basis m Good for Coulomb density, less so for exchange Best for pure (not hybrid) DFT Can be used in post-HF ab initio methods (MOLPRO, Turbomole, ORCA programs)

1 1 ij kl = ij m m kl r r m

(one of several possibilities)

Density Fitting / RI Methods: How to do itGAMESS-UK

Gaussian Keyword DensityFit; no choice of RI basis

NWChem specify cd basis (Coulomb) and optionally xc basis

MOLPRO

Local Correlation Methods In localised orbitals, probability of excitation decreases with spatial distance Massive reduction in scaling with system size Reduction in BSSE Not widely available (MOLPRO, Q-Chem)

CI in localised orbitals

Comparing Results from Different Programs Why dont I get the same result? Numerical accuracy Convergence criteria (SCF and geometry optimisation) DFT integration grid Optimisation may find different minimum

Cartesian/Spherical basis set Correlated/frozen core electrons in post-HF correlation methods Different definitions of B3LYP functional (at least two!)

Further InformationGaussian http://www.gaussian.com/g_ur/g03mantop.htm Foresman, Frisch: Exploring Chemistry with Electronic Structure Methods (available in library; I also have a copy)

NWChem http://www.emsl.pnl.gov/docs/nwchem/doc/user/index.html

GAMESS-UK http://www.cfs.dl.ac.uk/

MOLPRO http://www.molpro.net/

General Frank Jensen: Introduction to Computational Chemistry