Outline Escaping the Double Cone: Describing the Seam Space with Gateway Modes Columbus Worshop Argonne National Laboratory 005
Outline I. Motivation II. Perturbation Theory and Intersection Adapted Coordinates III. Determination of the Gateway Modes IV. Computational Schema V. Preliminary Results VI. Future Directions
Conical Intersections Their Role Recent studies have demonstrated that state of the art techniques for locating and characterizing conical intersections allow for the: Optimization of minimum energy intersections Elucidation of photodissociation pathways Inference of mechanistic processes based on routing through the branching plane The concept of routing is central to understanding dynamic processes involving multiple electronic states. The trajectories of molecules about a conical intersection are routed into pathways to correspond to the directions in which the degeneracy at the point of intersection is lifted in first order (i.e. g [energy difference], and h [interstate coupling] vectors).
Intersection Topography For eample, the double cone topography has been shown to be vital to interpreting photodissociation processes For H COH dissociation pathway: H COH + hν H CO + H D 0 = 1.3eV At left are the vectors corresponding to the +g direction +g +h +h +g Trajectories that come in at θ = 90, 70 are routed into the g direction g Yarony, J. Chem. Phys. 1 (005)
Dynamics About a Conical Intersection QUESTION: Is routing the whole story? Are there interactions involving coordinates outside the g-h plane that require eplicit consideration to achieve reliable results. To answer this question, a quantitative description of the coupling of the seam and branching coordinates is required. To be practical for an arbitrary molecule, the number of coupling coordinates must be reduced from N int to some smaller space. If possible to construct, such a coordinate system would be ideal for dynamics simulations in the vicinity of a conical intersection.
Hamiltonian Epansion Hamiltonian in CSF basis is epanded about a point of conical intersection through nd order and transformed into the crude adiabatic basis. 1 H CSF H l ( R)= ( R) + c R δ R H R δ R c R = E 0 ( R ) + h l 1 l δ R + δ R q δ R If the functions η are epanded through second order such that: η η Q P CSF CSF H ( R ) H ( R ) δr E ~ + + δ R H ( R ) δ R 0Q 1Q Q ( R) = η ( R ) + η ( R) + η ( R) ( R) ~ 1P ( R) ~ P = η + η ( R) ~ ~ Then it is possible to determine the first order term in the perturbation theory epansion to be: R CSF 0 ( R ) δ ( ) CSF ( ) l ( ) l + c R H δ Rc R 1 CSF l [ ] ( ) ( ) ( ) Q space includes states degenerate at R P space everything else [ ( ) ( (1) (0)( R )) I] ~ 0Q H ca1 E η = 0 where E K H ( ca 1) = h ( R ) δ R
Intersection Adapted Coordinates The optimal description of the vicinity around the conical intersection is given by intersection adapted coordinates. The branching space (i.e. space in which degeneracy is lifted linearly) is defined by only two coordinates ( and y), while the remaining coordinates (z i ) define the seam space: g ( R ) = ˆ = g g [ KK LL h ( R ) h ( R )] ( E ( R ) E ( R )) ( R ) ( R ) ŷ = = h h ( R ) ( R ) K L The seam coordinates: z (i), i = 1 - [N int ] are arbitrary and need only be orthogonal to g,h and each other. g h
Solution of 1 st Order Equation Since all first order interactions may be described in terms of the intersection adapted coordinates, the 1 st order Hamiltonian epression in new coordinate system. It will be useful to convert to cylindrical polar intersection adapted coordinates: ( i) ( i) ( ), sin( ), 1 ( int θ y = ρ θ z = z i = ) = ρ cos N Using these coordinates, the 1 st order Hamiltonian epression becomes simply: [ ( ca1) (1) H I] 0Q [ (1) I] 0Q 0 1 1 0 E η = gσ z + hyσ ε η = 0 where σ =, σ z = 1 0 0 1 This representation of the first order eigenvalue problem is conveniently solved using the transformation: where (1) H ~ = cos λ sin λ H sin λ cos λ ( ca1) cos λ sin λ sin λ cos λ = ρqσ z ( ) ( ) ( sin hsinθ q θ = g cosθ + h θ ) and tan λ = g cosθ
Hamiltonian Epansion nd Order Term Recalling that: where H () ( R ) q QQ H ( R ) ( ca) = + H δr QP ( R ) δr ( ) () 0Q (1) 1Q (0) Q [ H ca (0) IE ] η + [ h δr E I ] η + [ E ( R ) E I] η = 0 Unlie the first order epression, note that this equation involves contributions from both the Q and P spaces. H () ( PP ) 1 PQ H ( R ) (0) (0) ( R ) E ( R ) I E ( R ) the second order term in the partitioned Hamiltonian in the Q space is given by: K K It is not practical to compute such terms directly they will have to be obtained by fitting energy and derivative couplings to a functional form.
nd Order Epression Note H (ca) consists solely of terms that are quadratic in displacements from R i.e., y, y, z (), yz (), and z () z (l). Using polar coordinates we can liewise transform the second order piece of the Hamiltonian. where the A terms are second polynomials given by: ca) A = ρa R + w ( ca) ( ca) ( ca) ( R) = A I + Ag σ z Ah σ ( ca) H + ( ) ( ) ( ) ( c w w Aw The above polynomials are obtained by fitting the epression to ab initio data points. These terms can be interpreted such that: (ρ ) A w involves g-h plane interactions only composed of, yy, y interactions. (z) A w involves interactions between the g-h plane and the seam space composed of z, yz type interactions. (c) A defines the seam curvature and as such involves seam coordinates w only. R ( ) ( ) ( ) ρ ρ z = ( θ ) ( θ ) A R A + A ( c) w = w ( w) bl, l seam z A ( ) w z () l,z
nd Order Epression: In Detail These polynomials can be broen down into contributions from specific interactions: Branching plane only interactions (i.e., y (ρ ), y) are included in the : ( ρ, w) ( ρ, ) ( θ ) + a sin ( θ ) a cos( θ ) sin( θ ) ( ρ ) ( ρ, w) w w = a1 cos 3 A + A w Interactions involving one branching and one seam coordinate are given (z) by : A w where A ( z) w N int ( z ) ( θ,z) = A ( θ ) = 1 w z ( ) (1, w) (, ) ( θ ) = a cos( θ ) a sin( θ ) ( z ) w w A + ( 1, w) a (, w) a The and terms may be computed directly, or more practically, obtained by fits of ab initio data.
Derivative Coupling The epression for the derivative coupling in the ρ and z (i) directions is given by: f f z ρ = = Eamining this epression in detail, we see that the only terms that couple the g-h plane with seam coordinates are the ( z A ) and ( z. Recalling g A ) h the form of this epression: A A A ( z) w ( z ) ( z ) h cosλ + A q int N ( ) ( (1, w) (, w) θ,z = a cos( θ ) + a sin( θ )) = 1 We see that only four vectors: space seam space interactions. g a sin λ + (1, g ), a (, g ) l seam ( ρ ) ( ρ ) ( c) ( c) h cosλ + A q g sin λ A h, a ( ( h) ( l ) ( g ) ( l ) b z cosλ + b z sin λ) l cosλ + (1, h) ρ q, and A a g ρq z (, h) sin λ ( ) l account for all branching
Limit of Validity (c) A g In the event of seam curvature, seam space changes with displacements along the z () and the seam shifts to ρ 0 This may be quantified by decomposing δr into components in the g-h plane and perpendicular to it: δ R δ R + δ = (c) A h When the and curvature are near zero, the seam is piecewise linear; the seam eists for ρ = 0. Since displacements along any of the z () coordinates do not change the first order energy, care must be taen to ensure displacements δr do not have too large a component in any z () direction. R To ensure the validity of perturbation theory results, require: δ R δ R ρ
Gateway Modes: Motivation While the branching space is defined completely by the g-h coordinates, there remain N int modes perpindicular to the g-h plane that over which potentially significant interactions could be distributed. QUESTION: Is there a choice of coordinates that can consolidate these latent interactions into a space of reduced dimensionality? To answer this question, need to determine which terms lead to non-adiabatic transitions not focused in the g-h plane. These terms are found eclusive in : f z f z = ( (1, h) (, h) ) ( (1, g) (, g) a cosθ + a sinθ cosλ + a cosθ + a sinθ ) sin λ ( c), q The first term above does not vanish at the g-h plane (as does) and does not decay as 1/ρ lie the singular (1/ρ) ( dλ(θ)/dθ ) term. (c) f z + f z
Determining the Gateway Modes Now that the potentially important interactions that involve motion outside the g-h plane have been identified, coordinates to compactly describe these interactions are desired. This is accomplished by introducing a maimum of four new seam space coordinates ζ : int N ( j, w) ζ = ( j, w) ( ) a z = 1 Where j = 1, and w = g, h. The remaining N int 6coordinates are simply chosen to orthogonal to g,h and ζ. In summary: The and y vectors describe motion in the g-h plane, which is presumably dominant in the dynamics around a conical intersection. Potentially important interactions that involve motion out of the g-h plane are compactly described using the gateway mode coordinates The remaining N int 6 coordinates are presumed to be unimportant to the description of nuclear motion in the vicinity of a conical intersection.
Computational Schema In practice, the minimum amount of data required to compute all the A parameters are the energy difference gradients and derivative couplings at three points: 1. Use 1 energy difference gradients to obtain: () ( ρ, ) (,0) g E ρ = a1 4ρ ρ ρ z 1 () ( ρ, ) E (, ) h ρ π = a 4ρ ρ ρ z. Use derivative couplings to obtain: q q 0 ( ) ( ) ( ρ, g) π ρ π = f ρ, () ( ) ( ρ, h) fρ ρ,0 = a 1 a 1 E 0 g ( ) LK (1, ) ( ρ, ) = a 1 E h ( ) LK (, ) ( ρ, π ) = a ( ) ( ) (, g) π ρ π q f z ( ), = () ( ) ( 1, h) ρ q 0 f z ( ),0 = 3. Use previous results, as well as point at θ = π /4 to obtain a 3 terms : Parameters may be re-determined using the points θ π = 0,π/4,π/ and averaged. a ( ρ, g) ( ρ, g) ( ρ, [ a cos θ + a sin θ a cosθ sinθ] ) E cos λ + qfρ sin λ = 1 + 4ρ ρ ) ( ρ, h) ( ρ, h) E sin λ + qfρ cosλ = a1 cos θ + a sin θ + a 4ρ ρ 1 ( g) 3 ( ρ, [ cosθ sinθ] 1 ( h ) 3 a
Procedure To compute all the coupling parameters: 1. Find point of conical intersection using COLUMBUS and POLYHES.. Determine the set of points in a loop about the intersection employing a user define ρ and a set of angles θ in the g-h plane. 3. Compute the energy and gradient at each point in the loop [FIJ] and save the results in an accumulate.dat file. 4. Run the program FIJ to compute the derivative coupling parameters and potential terms involving g-h plane coordinates only. 5. Compute the energy and gradient at +/- points along each z (i) coordinate. 6. Determine the remaining potential terms that involve seam coordinates [MAKEPES]. z i z i ρ θ h h g g
Preliminary Results NH 3 Computing parameters at different value of ρ gives insight as to the effect of higher order terms. The magnitude of the differences in the computed parameters suggest small, but not negligible effect. For table defining the gateway coordinates, the inde (j,w) has been condensed to. ζ (1) is sufficient to represent ( 1, g) a, while arbitrary basis requires 3 z (i) cooridinates. While ζ (1) and ζ () are sufficient for (, h) a, 3 z (i) coordinates required. ρ 0.05 0.010 basis z ζ (, g) a ρ (, g ) 1 0.055 0.00338 0.0551 a ρ (, g a ρ ) (, h) 3 a ρ (, h) 1 a ρ (, h) a ρ 3 0.00935 0.00001 0.00046 0.00009 0.00048 0.00000 0.00019 0.057 0.03438 ( 1, g) (, g) ( 1, h) (, h) a a a a 1 0.04557 0.0000 0.00000 0.086 0.00000 0.00000 0.00000 0.00000 3 0.01851 0.00000 0.00000 0.0183 4 0.00071 0.00000 0.00000 0.00070 1 0.0490 0.00000 0.00000 0.03341 0.00000 0.00000 0.00000 0.0060 3 0.00000 0.00000 0.00000 0.00000 4 0.00000 0.00000 0.00000 0.00000
Preliminary Results NH 3 3 Most significant differences between g and ζ 1 vectors can be found in the nuclear motions of the N and H 1 atoms. Represents the second-order interaction that displays the largest coupling between a seam space and branching space coordinate. 1 0-1 - H 1 g/g ζ 1 N H H 3-3 -6-4 - 0 4
Preliminary Results H C OH Again, small deviations in the a parameters suggest a small, but non-negligible contribution from higher order terms. Larger set of z (i) terms more conclusively demonstrates the utility of the gateway mode description. Most of the a vectors have non-negligible contributions from 8 or more z (i), while the gateway representation requires only 3 or less coordinates. ρ 0.00 0.010 basis z ζ (, g) a ρ (, g ) 1 0.0349 0.00045 0.0311 a ρ (, g a ρ ) (, h) 3 a ρ (, h) 1 a ρ (, h) a ρ 3 0.00074 0.0016 0.00048 0.00000 0.00001 0.00041 0.00016 0.01881 0.01919 ( 1, g) (, g) ( 1, h) (, h) a a a a 1 0.0146 0.0005 0.00000 0.0081 0.0105 0.0001 0.00000 0.0058 3 0.00000 0.01198 0.0031 0.000 4 0.0513 0.00064 0.00000 0.00137 5 0.05084 0.0007 0.00000 0.0010 6 0.00001 0.00118 0.00154 0.00016 7 0.05103 0.00047 0.00000 0.00501 8 0.0098 0.0001 0.00000 0.00176 9 0.00069 0.00013 0.00000 0.0004 10 0.00084 0.0001 0.00000 0.00045 1 0.09045 0.00089 0.00000 0.00098 0.00000 0.0105 0.00334 0.0009 3 0.00000 0.00000 0.00011 0.00669 4 0.00000 0.00000 0.0013 0.00000
Preliminary Results H C OH.5 Significant differences between g and the ζ 1 vector are found mostly in the nuclear motion of C and O atoms. 1.5 0.5-0.5 H 1 g/g ζ 1 C 1 C O -1.5 H -.5 H 3-3.5-1.5 0.5.5 4.5
Future Directions: Vibrational Computations There is a fleibility in defining the basis for the a vibrational Hamiltonian. Other implementations have used functional forms that include: Distributed Gaussians Matri elements computed analytically or by quadrature. Harmonic oscillator Matri elements computed analytically, compact representation Our current Lanczos implementation utilizes distributed gaussian basis. Each mode as a linear combination of primitive gaussian functions positioned along the vibrational coordinate. Total size of the diagonalization problem is given by: N total bf N = modes i= 1 N bf ( mode ) i For a si mode molecule, i.e. HNCO, if number of gaussians per mode is 10, order of the resulting Hamiltonian matri is 10 6.
Lanczos Algorithm Given the size of the diagonalization problem AND that many roots are desired, a Lanczos procedure is used to compute the eigenvalues and eigenvectors iteratively. To diagonalize an N N matri A, the Lanczos algorithm defines a procedure to generate an N-length vector on the i th iteration of the process, q i, such that where The algorithm is given by: T = Q i T i AQ ( q, q, ) Q, i = 1 K q i Given r1 0; β1 r1 ; q0 = 0 For j = 1 to Niterations r q = i i βi ui = Aqi βiqi 1 α i = uiqi ri + 1 = ui αiqi β j+ 1 = ri + 1 i Matri-vector product Aq i is the major computation step The α i and β i are the diagonal of off-diagonal elements of the matri Τ ι, respectively.
Application of the Lanczos Solver The eigenvalues computed using the Lanczos algorithm are subject to a number of convergence properties The etremal eigenvalues in the eigenspectrum are the first to converge, with the largest eigenvalue converging near the same iteration as the smallest. In finite precision arithmetic, as the eigenvalues converge the Lanczos vectors necessarily begin to lost orthogonality. Loss of orthogonality gives rise to spurious roots (ghost eigenvalues). Orthogonality between the Lanczos vectors may be restored via one of a number of re-orthogonalization techniques requiring the storage of etra vectors. A Lanczos program, with a parallelized matri-vector multiplication step, has been written. Interface of the Lanczos solver with COLUMBUS, as well as the FIJ and MAKEPES modules is in progress and soon to be completed.
Summary Gateway modes provide a compact representation of second-order interactions between the branching and seam spaces that are predicted to be of the most significance in dynamical processes in the vicinity of a conical intersections. Practical epressions for the ey parameters that determine these modes have been determined. Their computation require only a small number of single point energy and gradient computations. Initial results are demonstrate the easy with which these coordinates may be incorporated into future dynamical studies.