Potential energy surfaces and applications for C m H n Potential energy surfaces and applications for C m H n Bastiaan J. Braams Emory University with Joel M. Bowman IAEA CRP meeting, Vienna, November 17–19, 2008
Potential energy surfaces and applications for C m H n Outline Introduction Sample applications Numerical approach Problems Outlook
Potential energy surfaces and applications for C m H n Introduction Potential energy surfaces Born-Oppenheimer approach: V = V ( X ) X : Collective (3 N ) nuclear coordinates. Also dipole moment surface (DMS) d ( X ), a vector quantity. Recent Bowman Group efforts: CH + 5 spectroscopy; CH 5 dissociation; C+C 2 H 2 reaction; C 2 H 3 , C 2 H + 3 , C 2 H + 5 spectroscopy; CH 3 CHO dissociation; HONO 2 , HOONO formation; CH 3 OH spectroscopy; CH 4 F dissociation; H + 5 , H + 4 quantum structure; 2 , H 5 O + C 3 H 4 O 2 spectroscopy; H 3 O − 2 spectroscopy; 2(H 2 O), 3(H 2 O) spectroscopy; 3(H 2 O)H + towards water+pH.
Potential energy surfaces and applications for C m H n Sample applications Malonaldehyde H-atom transfer
Potential energy surfaces and applications for C m H n Sample applications Saddle point for H-transfer
Potential energy surfaces and applications for C m H n 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 F 12 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 C m H n 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 C m H n 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 H 2 . Key lesson. Must use correct phase space sampling. Experimental branching ratio H to H 2 is 11:1; we get 14:1 from microcanonical sampling, 34:1 from standard mode sampling.
Potential energy surfaces and applications for C m H n Sample applications IR spectrum of CH 4 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 A ij . Comparison with HITRAN database for emissiion spectrum of CH 4 at 1000 K. Method can be applied to many molecules for which present database is less secure than for CH 4 . (Peter Hauschildt, Robert Warmbier)
Potential energy surfaces and applications for C m H n Numerical approach Choice of coordinates Considerations ◮ V is invariant under the point group symmetries: translation, rotation, reflection. Thus, 3 N − 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 ∈ R d , 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 C m H n Numerical approach Invariants and covariants Dipole moment: We use � d ( X ) = w i ( X ) r i ( X ) i Constraint: � i w i ( X ) = Z tot . Weights (effective charges) w i depend only on internal coordinates, like the PES V . PES is invariant; weights w i are covariant under nuclear permutations. Quadrupole moment, polarizability require further covariants.
Potential energy surfaces and applications for C m H n Numerical approach Invariants of finite groups - Introduction Easy case: Polynomials on R n invariant under Sym ( n ). Representation ( π p )( x ) = p ( π − 1 x ) for x ∈ R n ; ( π − 1 x ) i = x π i . Generated by the elementary monomials: � x k p k ( x ) = i i Every invariant polynomial f ( x ) has a unique representation in the form f ( x ) = poly ( p 1 ( x ) , . . . , p n ( x )). [Computational cost O (1) per term; compare with O ( n !) per term for symmetrized monomial basis.] Just as easy: Polynomials on R n 1 + ... + n K invariant under Sym ( n 1 ) × · · · Sym ( n K ) in the “natural” representation.
Potential energy surfaces and applications for C m H n 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 R n ): [Harm Derksen and Gregor Kemper, Computational Invariant Theory , Springer Verlag, 2002]. There exists a family of n primary generators , invariant polynomials p i (1 ≤ i ≤ n ), together with a family of secondary generators , invariant polynomials q α , such that every invariant polynomial f ( x ) has a unique representation in the form f ( x ) = � α poly α ( p 1 ( x ) , . . . , p n ( x )) q α ( x ).
Potential energy surfaces and applications for C m H n Numerical approach Invariants of finite groups - Example A: Case of G = Sym (2) acting on R 2 generated by reflections: ( x , y ) �→ ( x , − y ). May choose p 1 ( x , y ) = x , p 2 ( x , y ) = y 2 , and q 1 ( x , y ) = 1. Then: f ( x , y ) = poly 1 ( x , y 2 ) B: Case of G = Sym (2) acting on R 2 generated by inversions: ( x , y ) �→ ( − x , − y ). May choose p 1 ( x , y ) = x 2 , p 2 ( x , y ) = y 2 , and q 1 ( x , y ) = 1, q 2 ( x , y ) = xy . Then: f ( x , y ) = poly 1 ( x 2 , y 2 ) + poly 2 ( x 2 , y 2 ) xy
Potential energy surfaces and applications for C m H n 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 C m H n 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 C m H n 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 C m H n Numerical approach Numerical example Example, X5Y2 (H 5 O + 2 , H 5 C 2 , H 5 C + 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. � 28 � Without using symmetry, dimension , = 1,184,040. 7 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(H 2 O), H 4 C 3 O 2 , C 3 N 3 H 3 . Can anyway use many-body expansion.
Potential energy surfaces and applications for C m H n 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.
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