lecture 1: variational nuclear motion calculations lecture 2: … · 2006-02-22 · lecture 1:...
TRANSCRIPT
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)