Vibration-rotation spectra from first principles

Lecture 1: Variational nuclear motion calculationsLecture 2: Rotational motion, Spectra, PropertiesLecture 3: ApplicationsLecture 4: Calculations of spectroscopic accuracy

Jonathan TennysonDepartment of Physics and AstronomyUniversity College London

QUASAAR Winter School,Grenoble, Jan/Feb 2006

“(Variational calculations) will never displace the more traditional perturbation theory approach to calculating …..

vibration-rotation spectra”

Carter, Mills and Handy, J. Chem. Phys., 99, 4379 (1993)

Rotation-vibration energy levelsThe conventional view:

• Separate electronic and nuclear motion, The Born-Oppenheimer approximation

• Vibrations have small amplitude• Harmonic oscillations about equilibrium• Rotate as a rigid body• Rigid rotor model

Improved using perturbation theory

ButSmall amplitude vibrations often poor approximationWhat about dissociation?

Perturbation theory may not convergeDiverges for J > 7 for water

For high accuracy need electron-nuclear couplingImportant at the 1 cm−1 level for H-containing molecules

Equilibrium not always a useful conceptWhat about multiple minima?

Variational approaches: Ein > Ei

n+1

Internal coordinates: Eckart or Geometrically defined

• Exact nuclear kinetic energy operatorwithin the Born-Oppenheimer approximation

• Vibrational motion represented either byFinite Basis Representation (FBR) orGrid based Discrete Variable Representation (DVR)

• Solve problem using Variational Principle

• Potentials either ab initio or from fitting to spectra

Variational approaches

• Treats vibrations and rotations at the same time

• Interpret result in terms of potentials

• Only assume rigorous quantum numbers:

n, J, p, symmetry (eg ortho/para)

• Give spectra if dipole surface available

• Include all perturbations of energy levels and spectra

• Yield models that can be transferred between isotopomers

Provide a complete theoretical treatment with no assumptions

“Orthogonal” coordinate

Defined as ones in which KE operator is diagonal.Eg: Jacobi or Radau coordinatesMay not be chemically intuitiveMuch easier to program, can be more efficient (eg for a

DVR package)Use of several together gives “polyspherical” coordinates

Ιnternal coordinates:Orthogonal coordinates for triatomics

Orthogonal coordinates have diagonal kinetic energy operators. Important for DVR approached

Hamiltonians for nuclear motion

Laboratory fixed:3N coordinatesTranslation, vibration, rotationnot separately identified

Space fixed: remove translation of centre-of-mass3N−3 coordinatesVibration and rotation not separately identified

Body fixed: fix (“embed”) axis system in molecule3 rotational coordinates (2 also possible)3N−6 vibrational coordinates (or 3N−5)

Hamiltonians for nuclear motion

Laboratory fixed: Useless for variational calculations due to continuous translational “spectrum”.Used for Monte Carlo methods.

Space fixed:Requires choice of internal coordinates.Vibration and rotation not separately identified.Widely used for Van der Molecules.

Body fixed:Requires choice of internal axis system.Vibrational and rotational motion separately identified.Singularities!

New Hamiltonian for each coordinate/axis system

Same for J=0

Diatomic molecules: 1 vibrational mode

stretch

Hamiltonian:

Numerical solution: direct integration, trivial on a pc

Eg LEVEL by R J Le Roy, University of Waterloo Chemical Physics Research Report CP-642R (2001)

http://scienide.uwaterloo.ca/~leroy/level/

Triatomics: 3/4 vibrational mode

New mode: bend

Hamiltonian: many available, some generalNumerical solution: general programs available

3 degrees of freedom (4 for linear molecules)

Eg BOUND, DVR3D, TRIATOM see CCP6 program library http://www.ccp6.ac.ukor MORBID, DOPI

Tetratomics

New mode: umbrella

Hamiltonian: available for special cases eg “polyspherical” coordinate

Numerical solution: results for low energies

6 vibrational degrees of freedom

New mode: torsion

General polyspherical program: WAVR4 (in CPC program library)and some for special cases

Pentatomics

New modes:

book, ring puckering, wag, deformation, etc

Hamiltonian: for very few special cases eg XY4 systems, polyspherical coordinates

Numerical solution: very few (CH4)

12 degrees of freedom

Vibrating molecules with N atoms

Modes: all different types

Hamiltonian: not generally available but seeJ. Pesonen, Vibration-rotation kinetic energy operators: A geometricalgebra approach, J. Chem. Phys., 114, 10598 (2001).

Numerical solution: awaited for full problemBut MULTIMODE by S Carter & JM Bowman gives solutions for semi-rigid systems using SCF & CI methods plus approximationshttp://www.emory.edu/CHEMISTRY/faculty/bowman/multimode/

3N−6 degrees of freedom

Triatomics: general form of the Born-Oppenheimer Hamiltonian

KV vibrational kinetic energy operatorKVR vibration-rotation kinetic energy operator

(null if J=0)V the electronic potential energy surface

Steps in a calculation: choose…1. …a potential (determines accuracy)2. …coordinates (defines H)3. …basis functions for vibrational motion

Vibrational KE

Effective Hamiltonian after intergrationover angular and rotational coordinates.

Reduced masses (g1,g2) define coordinates

Vibrational KENon-orthogonal coordinates only

General coordinates

r2

r1θ

Choice of g1 and g2 defines coordinates

Basis functions.

Stretch functions:Morse oscillator (like)Harmonic oscillatorsSpherical oscillators, etc

Bending functions:Associate Legendre functionsJacobi polynomials

Rotational functions:Spherical top functions, DJ

MK

General functions:Floating spherical Gaussians Non-orthogonal

Must be complete setProblems as R 0

Coupling to rotational function ensures correct behaviour at linearity

Complete set of (2J+1) functions

Performing a Variational Calculation:(using a finite basis representation)

1. Construct individual matrix elements2. Construct full Hamiltonian matrix

3. Diagonalize Hamiltonian: get Ei and

Matrix elements

For general potential function, V,

need to obtain matrix elements using numerical quadratureFor Polynomial basis functions, Pn, useM-point Gaussian quadrature to givePoints, xi, Weights, wi

Can often obtain matrix elements overKinetic Energy operator analytically in closed form

Hnm = < n | T + V | m >

< n | V | m > = Σi wi Pn(xi) Pm(xi) V(xi)

Scales badly (~MN) with number of modes, N

Grid based methodsDiscrete Variable Representation (DVR) uses points and weights of Gaussian quadrature.Wavefunction obtained at grid of points, not as a continuous function.

DVR is isomorphic to an FBR

DVR versus FBRDVR advantages• Diagonal in the potential (quadrature approximation)

< α| V | β > = δαβ V(xα)• Sparse Hamiltonian matrix• Optimal truncation and diagonalization

based on adiabatic separation• Can select points to avoid singularities

DVR disadvantages• Not strictly variational (difficult to do small calculation)• Problems with coupled basis sets• Inefficient for non-orthogonal coordinate systems

Transformation between DVR and FBR quick & simple

Diagonalisation and Truncation in a DVR Eg waterStep 1: Lay down angular grid, Nγ points;Step 2: For each γi (ie fixed angle), set up and solve

the 2D radial problem;Step 3: Select all 2D solutions for which E < E2D

max;Step 4: Use selected 2D functions as basis for 3D problem;Step 5: Diagonalise 3D matrix;[Step 6: Back-transform wavefunction to original grid.]

Note: 1. Order important: coordinate with most grid points last;2. 3D diagonalisation rate determining so choice of E2D

maximportant;

3. Back-transformation needed if wavefunction required.

Matrix diagonalization• Matrices usually real symmetric• Diagonalization step rate limiting for triatomics, α N3.• Intermediate diagonalization and truncation

major aid to efficiency.

Iterative versus full matrix diagonalizer

• Is matrix sparse?• How many eigenvalues required?• Are eigenvectors needed?• Is matrix too large to store?

DVR calculations on a parallel processor

Aims: minimize inter-processor communication(and possibly input/output)

One strategy:Distribute 2D calculations across Nγ processors;Solve 2D problem.Compute 3D matrix (requires some communication);Diagonalise final 3D matrix

Choice of good parallel diagonaliser critical

Parallel runs of DVR3D (J Munro UCL)