Potential energy surfaces and applications for C m H n Bastiaan J. - - PowerPoint PPT Presentation

Potential energy surfaces and applications for CmHn

Potential energy surfaces and applications for CmHn

Bastiaan J. Braams Emory University with Joel M. Bowman IAEA CRP meeting, Vienna, November 17–19, 2008

Potential energy surfaces and applications for CmHn Outline

Introduction Sample applications Numerical approach Problems Outlook

Potential energy surfaces and applications for CmHn Introduction

Potential energy surfaces

Born-Oppenheimer approach: V = V (X) X: Collective (3N) nuclear coordinates. Also dipole moment surface (DMS) d(X), a vector quantity. Recent Bowman Group efforts: CH+

5 spectroscopy; CH5

dissociation; C+C2H2 reaction; C2H3, C2H+

3 , C2H+ 5 spectroscopy;

CH3CHO dissociation; HONO2, HOONO formation; CH3OH spectroscopy; CH4F dissociation; H+

5 , H+ 4 quantum structure;

C3H4O2 spectroscopy; H3O−

2 , H5O+ 2 spectroscopy; 2(H2O),

3(H2O) spectroscopy; 3(H2O)H+ towards water+pH.

Potential energy surfaces and applications for CmHn Sample applications

Malonaldehyde H-atom transfer

Potential energy surfaces and applications for CmHn Sample applications

Saddle point for H-transfer

Potential energy surfaces and applications for CmHn Sample applications

Tunneling splitting in malonaldehyde

[Yimin Wang, Bastiaan J. Braams, Joel M. Bowman, Stuart Carter, and David P. Tew, J. Chem. Phys. 128 (2008).] Ab initio by David Tew: CCSD(T) near basis set limit with F12

• correction. H-transfer barrier 4.1 kcal/mol.

Calculated splitting (DMC) for H: 22/cm; uncertainty 3/cm; measured 21.6/cm. For D: 3/cm; uncertainty 3/cm; measured 2.9/cm.

Potential energy surfaces and applications for CmHn Sample applications

Water dimer and trimer spectroscopy

[X. Huang, Bastiaan J. Braams, Joel M. Bowman, Ross

• E. A. Kelly, Jonathan Tennyson, Gerrit C. Groenenboom, and Ad

van der Avoird, J. Chem. Phys. 128 (2008).] [Yimin Wang, Stuart Carter, Bastiaan J. Braams, and Joel

• M. Bowman, J. Chem. Phys. 128 (2008).]

Dimer PES based on ≃ 30000 CCSD(T)/aug-cc-pvtz ab initio calculations, DMS based on MP2/aug-cc-pvtz. Vibration-Rotation-Tunneling splittings calculated in 6D QM. Multimode calculations of intramolecular frequencies.

Potential energy surfaces and applications for CmHn Sample applications

CH+

5 dissociative charge exchange

[Jennnifer E. Mann, Zhen Xie, John D. Savee, Bastiaan J. Braams, Joel M. Bowman, and Robert E. Continetti. JACS 130, 3730 (2008).] Charge exchange with Cs; measure and calculate kinetic energy distributions for products H and H2. Key lesson. Must use correct phase space sampling. Experimental branching ratio H to H2 is 11:1; we get 14:1 from microcanonical sampling, 34:1 from standard mode sampling.

Potential energy surfaces and applications for CmHn Sample applications

IR spectrum of CH4 for radiation transport modelling

[Robert Warmbier, Ralf Schneider, Amit Raj Sharma, Bastiaan

• J. Braams, Joel M. Bowman, and Peter H. Hauschildt. To be

published in Astronomy & Astrophysics.] Global PES and DMS for methane; then MULTIMODE calculations of ro-vibrational energy levels and dipole transition matrix methods; hence Einstein coefficients Aij. Comparison with HITRAN database for emissiion spectrum of CH4 at 1000 K. Method can be applied to many molecules for which present database is less secure than for CH4. (Peter Hauschildt, Robert Warmbier)

Potential energy surfaces and applications for CmHn Numerical approach

Choice of coordinates

Considerations

◮ V is invariant under the point group symmetries: translation,

rotation, reflection. Thus, 3N − 6 independent coordinates.

◮ V is invariant under permutations of like nuclei.

Use functions of the internuclear distances, r(i, j) = x(i) − x(j). For example, let y(i, j) = exp(−r(i, j)/λ); hence vector y ∈ Rd, d = N(N − 1)/2; and then V = p(y). Polynomial p must be invariant under permutations of like nuclei. Important earlier work: [J. N. Murrell et al., Molecular Potential Energy Functions, Wiley, 1984]. 3- and 4-atom systems.

Potential energy surfaces and applications for CmHn Numerical approach

Invariants and covariants

Dipole moment: We use d(X) =

• i

wi(X)ri(X) Constraint:

i wi(X) = Ztot.

Weights (effective charges) wi depend only on internal coordinates, like the PES V . PES is invariant; weights wi are covariant under nuclear permutations. Quadrupole moment, polarizability require further covariants.

Potential energy surfaces and applications for CmHn Numerical approach

Invariants of finite groups - Introduction

Easy case: Polynomials on Rn invariant under Sym(n). Representation (πp)(x) = p(π−1x) for x ∈ Rn; (π−1x)i = xπi. Generated by the elementary monomials: pk(x) =

• i

xk

i

Every invariant polynomial f (x) has a unique representation in the form f (x) = poly(p1(x), . . . , pn(x)). [Computational cost O(1) per term; compare with O(n!) per term for symmetrized monomial basis.] Just as easy: Polynomials on Rn1+...+nK invariant under Sym(n1) × · · · Sym(nK) in the “natural” representation.

Potential energy surfaces and applications for CmHn Numerical approach

Invariants of finite groups - General

Theory for the general case: invariant polynomials for a finite group G acting on a finite dimensional vector space (say Rn): [Harm Derksen and Gregor Kemper, Computational Invariant Theory, Springer Verlag, 2002]. There exists a family of n primary generators, invariant polynomials pi (1 ≤ i ≤ n), together with a family of secondary generators, invariant polynomials qα, such that every invariant polynomial f (x) has a unique representation in the form f (x) =

α polyα(p1(x), . . . , pn(x))qα(x).

Potential energy surfaces and applications for CmHn Numerical approach

Invariants of finite groups - Example

A: Case of G = Sym(2) acting on R2 generated by reflections: (x, y) → (x, −y). May choose p1(x, y) = x, p2(x, y) = y2, and q1(x, y) = 1. Then: f (x, y) = poly1(x, y2) B: Case of G = Sym(2) acting on R2 generated by inversions: (x, y) → (−x, −y). May choose p1(x, y) = x2, p2(x, y) = y2, and q1(x, y) = 1, q2(x, y) = xy. Then: f (x, y) = poly1(x2, y2) + poly2(x2, y2)xy

Potential energy surfaces and applications for CmHn Numerical approach

MAGMA computer algebra system

Developed at the University of Sydney, and elsewhere. Includes representation theory of finite groups.

◮ W. Bosma and J. Cannon: The Magma Handbook.

(20 chapters, ≃4000 pages.)

◮ Gregor Kemper and Allan Steel (1997) Some Algorithms in

Invariant Theory of Finite Groups. Use MAGMA to obtain primary and secondary invariants. Convert MAGMA output to Fortran code. Done for almost all molecular symmetry groups for at most 9 atoms.

Potential energy surfaces and applications for CmHn Numerical approach

MAGMA code

Fragment of Magma code.

intrinsic MolSymGen (nki::[RngIntElt]) -> GrpPerm {Permutation group for a Molecule} pairs:=[{i,j}:i in s[k],j in s[l],k in [1..l],l in [1..#nki]|i lt j] where s is [[&+nki[1..k-1]+1..&+nki[1..k]]:k in [1..#nki]]; return PermutationGroup<#pairs| [[Index(pairs,p^g):p in pairs]:g in GeneratorSequence(G)]> where G is DirectProduct([Sym(k):k in nki]); end intrinsic;

Potential energy surfaces and applications for CmHn Numerical approach

MAGMA output

Fragment of output for X5Y2. pv(205) = SYM d(i0,i1)*d(i0,j0)^5 pv(206) = SYM d(i0,i1)^4*d(i0,j0)*d(i1,j0) pv(207) = SYM d(i0,i1)^3*d(i0,i2)*d(i0,j0)*d(i1,j0) pv(208) = SYM d(i0,i1)^2*d(i0,i2)^2*d(i0,j0)*d(i1,j0) pv(209) = SYM d(i0,i1)*d(i0,i2)^3*d(i0,j0)*d(i1,j0) pv(210) = SYM d(i0,i2)^4*d(i0,j0)*d(i1,j0) pv(211) = SYM d(i0,i1)^2*d(i0,i2)*d(i1,i2)*d(i0,j0)*d(i1,j0) pv(212) = SYM d(i0,i1)*d(i0,i2)^2*d(i1,i2)*d(i0,j0)*d(i1,j0) pv(213) = SYM d(i0,i2)^3*d(i1,i2)*d(i0,j0)*d(i1,j0) pv(214) = SYM d(i0,i2)^2*d(i1,i2)^2*d(i0,j0)*d(i1,j0) pv(215) = SYM d(i0,i1)^2*d(i0,i2)*d(i0,i3)*d(i0,j0)*d(i1,j0) pv(216) = SYM d(i0,i1)*d(i0,i2)^2*d(i0,i3)*d(i0,j0)*d(i1,j0) pv(217) = SYM d(i0,i2)^3*d(i0,i3)*d(i0,j0)*d(i1,j0) pv(218) = SYM d(i0,i1)^2*d(i1,i2)*d(i0,i3)*d(i0,j0)*d(i1,j0)

Potential energy surfaces and applications for CmHn Numerical approach

Numerical example

Example, X5Y2 (H5O+

2 , H5C2, H5C+ 2 ); single Morse variable

expansion. N = 7, d = 21 (N(N − 1)/2); polynomials up to degree 7. Using symmetry, approximation space has dimension 8,717. Without using symmetry, dimension 28

7

• , = 1,184,040.

Least squares system: ∼ 50000 equations in ∼ 8717 unknowns. In addition, only the generators are costly. Can do larger problems using single expansion; even 9-atom systems with sufficient symmetry; 3(H2O), H4C3O2, C3N3H3. Can anyway use many-body expansion.

Potential energy surfaces and applications for CmHn Problems

Problems: It doesn’t scale

Larger molecules: PES for 3 atoms, many options. For 4 atoms, John Murrell, systematic expansion, 1970’s. For 5-9 atoms, our present work; also Mike Collins, Shepard interpolation. Global expansion in Morse variables not beyond ≃ 9 atoms. Many-body expansion may not converge; truncated MBE may not be accurate – it depends on the system.

Potential energy surfaces and applications for CmHn Problems

... It isn’t transferable

Every molecule has its own PES. (We use fragment data, not fragment potentials – for good reason, think CH2 singlet.) Need proper role for fragment potentials. Overlapping fragments; inclusion - exclusion counting; make it work also for reactions, charged systems, singlet and triplet surfaces.

Potential energy surfaces and applications for CmHn Problems

... It isn’t even global

Full-dimensional, yes. Beautiful for spectroscopy. Very good for dissociation experiments. Sufficiently global for small reactions. Reactions, large systems much more difficult. CH3COCH3 (Acetone) straight photo-dissociation we could do; CH3 + CH3CO reaction, I don’t know. The competition might be reparameterized semiempirical work.

Potential energy surfaces and applications for CmHn Problems

... It is barely well-posed

PES/DMS should fit ab initio, but ab initio may fail. We rely much on interpolation between fragment data and complex region data. Multireference CI as benchmark, but very expense for database.

Potential energy surfaces and applications for CmHn Problems

... It isn’t proper electronic structure

We have not addressed conical intersection issues. Single global fit

• ur style to the lower surface can’t be accurate.

Need to fit diabatic potential energy matrix W , but don’t know where to get the data. Need global data. W defined up to unitary transformation. May be covariant (not invariant) under permutations of nuclei. May even be multi-valued, if can be made single-valued by adding a coordinate.

Potential energy surfaces and applications for CmHn Outlook

Outlook

Big challenge: Water, including pH, flexible monomers. Needs divide-and-conquer; dipole and quadrupole moments, polarizability, and vdW forces; long-range electrostatics, local 2- and 3-monomer

• corrections. The ingredients are all there.

Previously indicated problems no real problem here.

◮ It will scale; MBE truncated at intrinsic 3-monomer term. ◮ Water is unique, no need to transfer. ◮ Ab initio calculations pose no problems, conical intersection

not an issue. Support: National Science Foundation, Department of Energy, Office of Naval Research.