Using the PGOPHER program to link experimental spectra with theory Rotations

Transcription

1 Using the PGOPHER program to link experimental spectra with theory Rotations Colin M. Western chool of Chemistry, University of Bristol, Bristol B8 1T, UK C.M.Western@bristol.ac.uk

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3 What does PGOPHER do? Two distinct modes imulates rotation structure of molecular spectra: inear molecules, symmetric and asymmetric tops Includes many effects (electron and nuclear spin, electric and magnetic fields). imulate many interacting vibronic states simultaneously, including interactions between states tandard rotational Hamiltonians Vibronic structure associated with an electronic state Calculate interacting states, Renner-Teller Jahn-Teller Franck-Condon factors Force Field calculations Based on harmonic basis set

4 Undergraduate practical HCl IR spectrum ow resolution IR spectrum from NIT website in JDX format Demonstrates basic interactive fitting taking line positions from spectrum hort lines join obs to calc Experiment imulation Right click and drag here to put observed position here Right click here to display line info here

5 Interactive Adjustment of Parameters Right Click Roll mouse wheel

6 Adding another isotope H 5 Cl and H 7 Cl tarting from the previous data, a simulation showing both isotopes is easy to generate by copying and pasting the Molecule object The constants and name can then be changed as required; here the abundance has been set to 1/ and a small shift applied to the band origin

7 The Experimentalist s Approach Fitting of spectra need a model that reproduces the observed pattern (position, intensities) of transitions Typically use empirical model (easy to calculate) Parameters in model not necessarily exactly the same as theoretical equivalent Consider rotational constant: E(J) = BJ(J+1), B = h / (8π μr ) Truncation: E(J) = BJ(J+1) DJ (J+1) + Vibrational Averaging: B v = B e α(v+1/) + Mixing of other electronic states: J + Program I have developed designed to help experimentalists extract information from spectra

8 How does it work? Energy levels imple basis set expansion: Typically small number of basis states, ~J Hamiltonian expressed in terms of angular momentum operators: inear: c 1 c a 1 ymmetric top: Asymmetric top: H ψ = E ψ then reduces to finding eigenvalues of a matrix Intensities require the coefficients from the diagonalisation: H B R H B N H A N A C B N z B N C N a b c all space * f μ i d f, i c f c i f i

9 pectrum of the A P v'=1 X v''=0 transition in O 1+1 MPI

10 Which terms are needed? O A P v'=7 X v''=0 Ω' = Experiment imulation (All terms) imulation X: B, λ A: Origin, B, A Wavenumber/cm -1

11 Hund's Case (a) J,,,, W good quantum numbers Appropriate for moderate spin-orbit coupling (A) Most molecules will be a mixture of Hund's cases. PGOPHER uses Hund s case (a) basis set Hund s case (b) (inset) calculated exactly as basis set complete ; N JM R J R J N z, W J N W 1 N 1 ; ; JM W W N W

12 Open hell inear Molecule Hamiltonian H J R ) ( A B A B A (Diagonal) pin-orbit (Diagonal) Rotation uncoupling Off diagonal in electronic state 1) ( 1) ( W J J B J J B B A 1 Homogeneous tate Mixing Uncoupling Constant J J B x y B No general solution to the above uncoupling is the key term giving a complicated result Must be included unless A is large (A >> BJ)

13 Complications pin Orbit plitting AΛΣ term predicts equally spaced levels but: 149 cm cm 1 Difference conventionally ascribed to (electron) spin-spin interaction: Formally due to interaction between the magnetic dipoles of the unpaired electrons Also gives Σ 0 Σ 1 splitting z

14 Complications pin Orbit plitting II is not quite right for the form of the spin-orbit operator To see why, express angular momenta as sum from individual electrons: so: But true form of spin orbit operator is: o simple form is not exactly right Extra term is pin- other orbit operator True form of operator includes term that mixes states with different Gives rise to spin forbidden transitions. A 1 l l 1 s s s s s s s s l l l l l l 1 1 s a s a l l 1 1 s a s a l l

15 Effect of Neglected terms in the Hamiltonian Mixes states together so forbidden transitions allowed: 1 T 1 1 T1 Allowed 0 Forbidden Add Mixing Allowed Energy level shifts; second order perturbation theory suggests: E i j i h E If (say) matrix elements then overall effect of a distant z electronic state (A and BJ(J+1) << E i E j ) is a shift Indistinguishable from spin-spin term, so is an z effective Hamiltonian; also changed many other terms (A, B) Above is van Vleck perturbation theory or contact transformation i j j E h j i 0 (Weakly) Allowed z

16 ocal Perturbations in O If two levels of the same symmetry are close in energy, the levels can mix and push apart: No Mixing V = V = V = 1 V = With Mixing V = 0 tate V = 1 tate 1 V = 0

17 Origin of ocal Perturbations High vibrational levels of, say, ground state can interact weakly with excited electronic states cm A Π O X Σ R/Å

18 Perturbations in O A P X 7 0 Transition shows a perturbation by a dark state localised to just a few levels of one parity Dark state is shown in red in simulation and energy level plot ymbols on (reduced) energy level plot are observed energy levels generated from a line list assuming lower state levels are exact

19 tructure of Constants View Overall name = File Name imulation parameters e.g. T, width Molecule Isotopomer could add 4 O Manifolds = group of vibronic states; states must be in same Manifold to interact Vibronic tates Perturbation Transition Moment

20 Energy 0.5J(J+1) (Reduced) Energy evel Plot Note energy level plot reduced by subtracting BJ(J+1) Π Π 1 Perturbing tate (Probably high v' of lower state) Π 0

21 Assigning Quantum Numbers elect all states of a particular symmetry: J = 5, parity Π 0 Hamiltonian Matrix Π 1 Π Perturbing tate Perturbation Matrix Element Π 0 Π 1 Perturbing tate Π Energies

22 Assigning Quantum Numbers II If A = 14 cm 1 (rather than 140) much more difficult to assign quantum numbers Classic Hund s case (a) (b) switch as a function of J: J = 5 J = 50 Hund s case (b) basis would be better as J = 50 but worse at J = 5 (Basis is complete, so makes no difference to energies or intensities)

23 Algorithm for Assigning Quantum Numbers Hamiltonian matrix blocked to avoid accidental mixing For each symmetry, to assign vibronic state, sum contribution from all basis states for that vibronic state and choose n largest. Repeat for each vibronic state Each vibronic state has a standard energy order for the n basis states: For linear molecules Π > Π 1 > Π 0 if A > 0 (Regular) (Regular/Inverted test actually on diagonal matrix elements for J=) Gives quantum number assignments consistent with literature in almost all cases, but strong vibronic state mixing can still give problems. Only J, symmetry and eigenvalue number guaranteed.

24 tandard Order of Basis tates inear Molecules: A > 0 A < 0 Highest E Π Π 0 F N = J + 1 Π 1 Π 1 F N = J owest E Π 0 Π F 1 N = J 1 ymmetric tops: K = 0, 1,, (Prolate) K =, 1, 0 (Oblate) Asymmetric tops: Order always Energy increases with K a, decreases with K c

25 ymbolic Matrix Elements PGOPHER can display matrix elements for in symbolic format by duplicating the calculation using a basic symbolic algebra system Perturbation is <Dark J+- v=7>: <J,ambda=0,Omega=0 J+- J,ambda=-1,Omega=-1>= -sqrt(j*(j+1)) <J,ambda=0,Omega=0 J+- J,ambda=1,Omega=1> = sqrt(j*(j+1)) B is <J,ambda=1,Omega=0 B J,ambda=1,Omega=0> = J*(J+1)+ cf general formula: B B J ( J 1) ( 1) W J ( J 1) 0 1 ee documentation on RquaredH flag for discussion