Outline. Escaping the Double Cone: Describing the Seam Space with Gateway Modes. Columbus Workshop Argonne National Laboratory 2005
|
|
- Shawn Lawrence
- 5 years ago
- Views:
Transcription
1 Outline Escaping the Double Cone: Describing the Seam Space with Gateway Modes Columbus Worshop Argonne National Laboratory 005
2 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
3 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).
4 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)
5 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.
6 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
7 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
8 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 E η = gσ z + hyσ ε η = 0 where σ =, σ z = 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
9 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.
10 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
11 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.
12 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
13 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
14 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
15 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.
16 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
17 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
18 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 basis z ζ (, g) a ρ (, g ) a ρ (, g a ρ ) (, h) 3 a ρ (, h) 1 a ρ (, h) a ρ ( 1, g) (, g) ( 1, h) (, h) a a a a
19 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 H 1 g/g ζ 1 N H H
20 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 basis z ζ (, g) a ρ (, g ) a ρ (, g a ρ ) (, h) 3 a ρ (, h) 1 a ρ (, h) a ρ ( 1, g) (, g) ( 1, h) (, h) a a a a
21 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 H 1 g/g ζ 1 C 1 C O -1.5 H -.5 H
22 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.
23 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.
24 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.
25 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.
Conical Intersections. Spiridoula Matsika
Conical Intersections Spiridoula Matsika The Born-Oppenheimer approximation Energy TS Nuclear coordinate R ν The study of chemical systems is based on the separation of nuclear and electronic motion The
More informationPROBLEMS In each of Problems 1 through 12:
6.5 Impulse Functions 33 which is the formal solution of the given problem. It is also possible to write y in the form 0, t < 5, y = 5 e (t 5/ sin 5 (t 5, t 5. ( The graph of Eq. ( is shown in Figure 6.5.3.
More informationConical Intersections: Electronic Structure,Dynamics and Spectroscopy, Yarkony eds. World Scientific Publishing, Singapore, (2004) 1/20/2006 1
Conical Intersections: Electronic Structure,Dynamics and Spectroscopy, Wolfgang Domcke,, Horst Köppel, K and David R. Yarkony eds. World Scientific Publishing, Singapore, (2004) 1/20/2006 1 Characterizing
More informationName of the Student:
Engineering Mathematics 016 SUBJECT NAME : Engineering Mathematics - I SUBJECT CODE : MA111 MATERIAL NAME : Universit Questions REGULATION : R008 WEBSITE : wwwhariganeshcom UPDATED ON : Januar 016 TEXTBOOK
More informationMATHEMATICS 200 April 2010 Final Exam Solutions
MATHEMATICS April Final Eam Solutions. (a) A surface z(, y) is defined by zy y + ln(yz). (i) Compute z, z y (ii) Evaluate z and z y in terms of, y, z. at (, y, z) (,, /). (b) A surface z f(, y) has derivatives
More informationSupporting Information. I. A refined two-state diabatic potential matrix
Signatures of a Conical Intersection in Adiabatic Dissociation on the Ground Electronic State Changjian Xie, Christopher L. Malbon, # David R. Yarkony, #,* Daiqian Xie,,%,* and Hua Guo,* Department of
More informationChapter 6. Nonlinear Equations. 6.1 The Problem of Nonlinear Root-finding. 6.2 Rate of Convergence
Chapter 6 Nonlinear Equations 6. The Problem of Nonlinear Root-finding In this module we consider the problem of using numerical techniques to find the roots of nonlinear equations, f () =. Initially we
More informationHarmonic Oscillator Eigenvalues and Eigenfunctions
Chemistry 46 Fall 217 Dr. Jean M. Standard October 4, 217 Harmonic Oscillator Eigenvalues and Eigenfunctions The Quantum Mechanical Harmonic Oscillator The quantum mechanical harmonic oscillator in one
More informationMathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes
Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.
More informationMATH section 3.1 Maximum and Minimum Values Page 1 of 7
MATH section. Maimum and Minimum Values Page of 7 Definition : Let c be a number in the domain D of a function f. Then c ) is the Absolute maimum value of f on D if ) c f() for all in D. Absolute minimum
More informationR-Linear Convergence of Limited Memory Steepest Descent
R-Linear Convergence of Limited Memory Steepest Descent Frank E. Curtis, Lehigh University joint work with Wei Guo, Lehigh University OP17 Vancouver, British Columbia, Canada 24 May 2017 R-Linear Convergence
More informationGiven the vectors u, v, w and real numbers α, β, γ. Calculate vector a, which is equal to the linear combination α u + β v + γ w.
Selected problems from the tetbook J. Neustupa, S. Kračmar: Sbírka příkladů z Matematiky I Problems in Mathematics I I. LINEAR ALGEBRA I.. Vectors, vector spaces Given the vectors u, v, w and real numbers
More information33A Linear Algebra and Applications: Practice Final Exam - Solutions
33A Linear Algebra and Applications: Practice Final Eam - Solutions Question Consider a plane V in R 3 with a basis given by v = and v =. Suppose, y are both in V. (a) [3 points] If [ ] B =, find. (b)
More informationLines and points. Lines and points
omogeneous coordinates in the plane Homogeneous coordinates in the plane A line in the plane a + by + c is represented as (a, b, c). A line is a subset of points in the plane. All vectors (ka, kb, kc)
More informationReview of Matrices and Vectors 1/45
Reiew of Matrices and Vectors /45 /45 Definition of Vector: A collection of comple or real numbers, generally put in a column [ ] T "! Transpose + + + b a b a b b a a " " " b a b a Definition of Vector
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 71 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationCalculus of Variation An Introduction To Isoperimetric Problems
Calculus of Variation An Introduction To Isoperimetric Problems Kevin Wang The University of Sydney SSP Working Seminars, MATH2916 May 4, 2013 Contents I Lagrange Multipliers 2 1 Single Constraint Lagrange
More information3.3.1 Linear functions yet again and dot product In 2D, a homogenous linear scalar function takes the general form:
3.3 Gradient Vector and Jacobian Matri 3 3.3 Gradient Vector and Jacobian Matri Overview: Differentiable functions have a local linear approimation. Near a given point, local changes are determined by
More informationNational Quali cations
National Quali cations AH08 X747/77/ Mathematics THURSDAY, MAY 9:00 AM :00 NOON Total marks 00 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions which contain
More informationExample - Newton-Raphson Method
Eample - Newton-Raphson Method We now consider the following eample: minimize f( 3 3 + -- 4 4 Since f ( 3 2 + 3 3 and f ( 6 + 9 2 we form the following iteration: + n 3 ( n 3 3( n 2 ------------------------------------
More information1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates...
Contents 1 Geometry of R 2 2 1.1 Conic Sections............................................ 2 1.2 Parametric Equations........................................ 3 1.3 More Parametric Equations.....................................
More informationPhys 622 Problems Chapter 5
1 Phys 622 Problems Chapter 5 Problem 1 The correct basis set of perturbation theory Consider the relativistic correction to the electron-nucleus interaction H LS = α L S, also known as the spin-orbit
More informationSteady and unsteady diffusion
Chapter 5 Steady and unsteady diffusion In this chapter, we solve the diffusion and forced convection equations, in which it is necessary to evaluate the temperature or concentration fields when the velocity
More information3 x x+1. (4) One might be tempted to do what is done with fractions, i.e., combine them by means of a common denominator.
Lecture 6 Techniques of integration (cont d) Integration of rational functions by partial fractions Relevant section from Stewart, Eighth Edition: 7.4 In this section, we consider the integration of rational
More informationUseful Mathematics. 1. Multivariable Calculus. 1.1 Taylor s Theorem. Monday, 13 May 2013
Useful Mathematics Monday, 13 May 013 Physics 111 In recent years I have observed a reticence among a subpopulation of students to dive into mathematics when the occasion arises in theoretical mechanics
More informationTrig. Past Papers Unit 2 Outcome 3
PSf Written Questions Trig. Past Papers Unit utcome 3 1. Solve the equation 3 cos + cos = 1 in the interval 0 360. 5 Part Marks Level Calc. Content Answer U C3 5 A/B CR T10 60, 131 8, 8, 300 000 P Q5 1
More informationabc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS
More informationNumerical methods to compute optical errors due to stress birefringence
Numerical methods to compute optical errors due to stress birefringence Keith B. Doyle Optical Research Associates, 8 West Park Drive, Westborough, MA Victor L. Genberg & Gregory J. Michels Sigmadyne,
More informationNonlinear Oscillations and Chaos
CHAPTER 4 Nonlinear Oscillations and Chaos 4-. l l = l + d s d d l l = l + d m θ m (a) (b) (c) The unetended length of each spring is, as shown in (a). In order to attach the mass m, each spring must be
More informationNumerical Methods. Root Finding
Numerical Methods Solving Non Linear 1-Dimensional Equations Root Finding Given a real valued function f of one variable (say ), the idea is to find an such that: f() 0 1 Root Finding Eamples Find real
More information5.3 The Power Method Approximation of the Eigenvalue of Largest Module
192 5 Approximation of Eigenvalues and Eigenvectors 5.3 The Power Method The power method is very good at approximating the extremal eigenvalues of the matrix, that is, the eigenvalues having largest and
More informationUNCONSTRAINED OPTIMIZATION PAUL SCHRIMPF OCTOBER 24, 2013
PAUL SCHRIMPF OCTOBER 24, 213 UNIVERSITY OF BRITISH COLUMBIA ECONOMICS 26 Today s lecture is about unconstrained optimization. If you re following along in the syllabus, you ll notice that we ve skipped
More informationabc Mathematics Further Pure General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES
abc General Certificate of Education Mathematics Further Pure SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER
More information5.80 Small-Molecule Spectroscopy and Dynamics
MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.80 Lecture
More informationVARIATIONAL PRINCIPLES
CHAPTER - II VARIATIONAL PRINCIPLES Unit : Euler-Lagranges s Differential Equations: Introduction: We have seen that co-ordinates are the tools in the hands of a mathematician. With the help of these co-ordinates
More information4Divergenceandcurl. D ds = ρdv. S
4Divergenceandcurl Epressing the total charge Q V contained in a volume V as a 3D volume integral of charge density ρ(r), wecanwritegauss s law eamined during the last few lectures in the general form
More informationSYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS
SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be one -hour paper consisting of 4 questions..
More informationand Rational Functions
chapter This detail from The School of Athens (painted by Raphael around 1510) depicts Euclid eplaining geometry. Linear, Quadratic, Polynomial, and Rational Functions In this chapter we focus on four
More informationFinding normalized and modularity cuts by spectral clustering. Ljubjana 2010, October
Finding normalized and modularity cuts by spectral clustering Marianna Bolla Institute of Mathematics Budapest University of Technology and Economics marib@math.bme.hu Ljubjana 2010, October Outline Find
More informationThe COLUMBUS Project - General Purpose Ab Initio Quantum Chemistry. I. Background and Overview
The COLUMBUS Project - General Purpose Ab Initio Quantum Chemistry I. Background and Overview Ron Shepard Chemistry Division Argonne National Laboratory CScADS Workshop, Snowbird, Utah, July 23, 2007 Quantum
More informationIntroduction to Geometry Optimization. Computational Chemistry lab 2009
Introduction to Geometry Optimization Computational Chemistry lab 9 Determination of the molecule configuration H H Diatomic molecule determine the interatomic distance H O H Triatomic molecule determine
More informationAssignment 6. Using the result for the Lagrangian for a double pendulum in Problem 1.22, we get
Assignment 6 Goldstein 6.4 Obtain the normal modes of vibration for the double pendulum shown in Figure.4, assuming equal lengths, but not equal masses. Show that when the lower mass is small compared
More informationFunctions with orthogonal Hessian
Functions with orthogonal Hessian B. Dacorogna P. Marcellini E. Paolini Abstract A Dirichlet problem for orthogonal Hessians in two dimensions is eplicitly solved, by characterizing all piecewise C 2 functions
More information13-2 Text: 28-30; AB: 1.3.3, 3.2.3, 3.4.2, 3.5, 3.6.2; GvL Eigen2
The QR algorithm The most common method for solving small (dense) eigenvalue problems. The basic algorithm: QR without shifts 1. Until Convergence Do: 2. Compute the QR factorization A = QR 3. Set A :=
More informationSpectral Processing. Misha Kazhdan
Spectral Processing Misha Kazhdan [Taubin, 1995] A Signal Processing Approach to Fair Surface Design [Desbrun, et al., 1999] Implicit Fairing of Arbitrary Meshes [Vallet and Levy, 2008] Spectral Geometry
More information2-5 The Calculus of Scalar and Vector Fields (pp.33-55)
9/1/ sec _5 empty.doc 1/9-5 The Calculus of Scalar and Vector Fields (pp.33-55) Q: A: 1... 5. 3. 6. A. The Integration of Scalar and Vector Fields 1. The Line Integral 9/1/ sec _5 empty.doc /9 Q1: A C
More informationSecond-order cone programming
Outline Second-order cone programming, PhD Lehigh University Department of Industrial and Systems Engineering February 10, 2009 Outline 1 Basic properties Spectral decomposition The cone of squares The
More informationEXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants
EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For
More informationNational Quali cations AHEXEMPLAR PAPER ONLY
National Quali cations AHEXEMPLAR PAPER ONLY EP/AH/0 Mathematics Date Not applicable Duration hours Total marks 00 Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions
More informationMOL 410/510: Introduction to Biological Dynamics Fall 2012 Problem Set #4, Nonlinear Dynamical Systems (due 10/19/2012) 6 MUST DO Questions, 1
MOL 410/510: Introduction to Biological Dynamics Fall 2012 Problem Set #4, Nonlinear Dynamical Systems (due 10/19/2012) 6 MUST DO Questions, 1 OPTIONAL question 1. Below, several phase portraits are shown.
More informationGraded Questions on Matrices. 1. Reduce the following matrix to its normal form & hence find its rank Where
Graded Questions on Matrices MATRICES Rank of Matri : Normal Form 1. Reduce the following matri to its normal form & hence find its rank A N M 2 3 1 1 1 1 2 4 3 1 3 2 6 3 0 7 Q P [Dec. 07, May 10] 2. Reduce
More informationAnd similarly in the other directions, so the overall result is expressed compactly as,
SQEP Tutorial Session 5: T7S0 Relates to Knowledge & Skills.5,.8 Last Update: //3 Force on an element of area; Definition of principal stresses and strains; Definition of Tresca and Mises equivalent stresses;
More informationPractical and Efficient Evaluation of Inverse Functions
J. C. HAYEN ORMATYC 017 TEXT PAGE A-1 Practical and Efficient Evaluation of Inverse Functions Jeffrey C. Hayen Oregon Institute of Technology (Jeffrey.Hayen@oit.edu) ORMATYC 017 J. C. HAYEN ORMATYC 017
More informationUniversity of Alabama Department of Physics and Astronomy. PH 125 / LeClair Spring A Short Math Guide. Cartesian (x, y) Polar (r, θ)
University of Alabama Department of Physics and Astronomy PH 125 / LeClair Spring 2009 A Short Math Guide 1 Definition of coordinates Relationship between 2D cartesian (, y) and polar (r, θ) coordinates.
More informationNon-degenerate Perturbation Theory. and where one knows the eigenfunctions and eigenvalues of
on-degenerate Perturbation Theory Suppose one wants to solve the eigenvalue problem ĤΦ = Φ where µ =,1,2,, E µ µ µ and where Ĥ can be written as the sum of two terms, ˆ ˆ ˆ ˆ ˆ ˆ H = H + ( H H ) = H +
More informationBohr & Wheeler Fission Theory Calculation 4 March 2009
Bohr & Wheeler Fission Theory Calculation 4 March 9 () Introduction The goal here is to reproduce the calculation of the limiting Z /A against spontaneous fission Z A lim a S. (.) a C as first done by
More informationThe Gradient. Consider the topography of the Earth s surface.
9/16/5 The Gradient.doc 1/8 The Gradient Consider the topography of the Earth s surface. We use contours of constant elevation called topographic contours to epress on maps (a -dimensional graphic) the
More informationStatistical Geometry Processing Winter Semester 2011/2012
Statistical Geometry Processing Winter Semester 2011/2012 Linear Algebra, Function Spaces & Inverse Problems Vector and Function Spaces 3 Vectors vectors are arrows in space classically: 2 or 3 dim. Euclidian
More informationAnswer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
More informationPage 404. Lecture 22: Simple Harmonic Oscillator: Energy Basis Date Given: 2008/11/19 Date Revised: 2008/11/19
Page 404 Lecture : Simple Harmonic Oscillator: Energy Basis Date Given: 008/11/19 Date Revised: 008/11/19 Coordinate Basis Section 6. The One-Dimensional Simple Harmonic Oscillator: Coordinate Basis Page
More informationAdvanced Higher Grade
Practice Eamination A (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours 0 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working..
More informationLecture 1: Systems of linear equations and their solutions
Lecture 1: Systems of linear equations and their solutions Course overview Topics to be covered this semester: Systems of linear equations and Gaussian elimination: Solving linear equations and applications
More informationEngineering Mathematics 2018 : MA6151
Engineering Mathematics 08 NAME OF THE SUBJECT : Mathematics I SUBJECT CODE : MA65 NAME OF THE METERIAL : Part A questions REGULATION : R 03 WEBSITE : wwwhariganeshcom UPDATED ON : November 07 TEXT BOOK
More informationNumerical Methods. Elena loli Piccolomini. Civil Engeneering. piccolom. Metodi Numerici M p. 1/??
Metodi Numerici M p. 1/?? Numerical Methods Elena loli Piccolomini Civil Engeneering http://www.dm.unibo.it/ piccolom elena.loli@unibo.it Metodi Numerici M p. 2/?? Least Squares Data Fitting Measurement
More informationEngineering Mathematics 2018 : MA6151
Engineering Mathematics 08 NAME OF THE SUBJECT : Mathematics I SUBJECT CODE : MA65 MATERIAL NAME : Universit Questions REGULATION : R 03 WEBSITE : wwwhariganeshcom UPDATED ON : November 07 TEXT BOOK FOR
More information5.4 Given the basis e 1, e 2 write the matrices that represent the unitary transformations corresponding to the following changes of basis:
5 Representations 5.3 Given a three-dimensional Hilbert space, consider the two observables ξ and η that, with respect to the basis 1, 2, 3, arerepresentedby the matrices: ξ ξ 1 0 0 0 ξ 1 0 0 0 ξ 3, ξ
More information1. Which of the following defines a function f for which f ( x) = f( x) 2. ln(4 2 x) < 0 if and only if
. Which of the following defines a function f for which f ( ) = f( )? a. f ( ) = + 4 b. f ( ) = sin( ) f ( ) = cos( ) f ( ) = e f ( ) = log. ln(4 ) < 0 if and only if a. < b. < < < < > >. If f ( ) = (
More information( ) = 9φ 1, ( ) = 4φ 2.
Chemistry 46 Dr Jean M Standard Homework Problem Set 6 Solutions The Hermitian operator A ˆ is associated with the physical observable A Two of the eigenfunctions of A ˆ are and These eigenfunctions are
More informationGeometric Modeling Summer Semester 2010 Mathematical Tools (1)
Geometric Modeling Summer Semester 2010 Mathematical Tools (1) Recap: Linear Algebra Today... Topics: Mathematical Background Linear algebra Analysis & differential geometry Numerical techniques Geometric
More informationACS MATHEMATICS GRADE 10 WARM UP EXERCISES FOR IB HIGHER LEVEL MATHEMATICS
ACS MATHEMATICS GRADE 0 WARM UP EXERCISES FOR IB HIGHER LEVEL MATHEMATICS DO AS MANY OF THESE AS POSSIBLE BEFORE THE START OF YOUR FIRST YEAR IB HIGHER LEVEL MATH CLASS NEXT SEPTEMBER Write as a single
More informationSingle Particle Motion
Single Particle Motion C ontents Uniform E and B E = - guiding centers Definition of guiding center E gravitation Non Uniform B 'grad B' drift, B B Curvature drift Grad -B drift, B B invariance of µ. Magnetic
More informationMath Subject GRE Questions
Math Subject GRE Questions Calculus and Differential Equations 1. If f() = e e, then [f ()] 2 [f()] 2 equals (a) 4 (b) 4e 2 (c) 2e (d) 2 (e) 2e 2. An integrating factor for the ordinary differential equation
More informationNormal Mode Decomposition of 2N 2N symplectic matrices
David Rubin February 8, 008 Normal Mode Decomposition of N N symplectic matrices Normal mode decomposition of a 4X4 symplectic matrix is a standard technique for analyzing transverse coupling in a storage
More informationChemistry 431. NC State University. Lecture 17. Vibrational Spectroscopy
Chemistry 43 Lecture 7 Vibrational Spectroscopy NC State University The Dipole Moment Expansion The permanent dipole moment of a molecule oscillates about an equilibrium value as the molecule vibrates.
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED GCE UNIT / MATHEMATICS (MEI Further Methods for Advanced Mathematics (FP THURSDAY JUNE Additional materials: Answer booklet (8 pages Graph paper MEI Eamination Formulae and Tables (MF Morning
More informationOmm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics
LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector
More informationComputations with Discontinuous Basis Functions
Computations with Discontinuous Basis Functions Carl Sovinec University of Wisconsin-Madison NIMROD Team Meeting November 12, 2011 Salt Lake City, Utah Motivation The objective of this work is to make
More informationA2 Assignment zeta Cover Sheet. C3 Differentiation all methods. C3 Sketch and find range. C3 Integration by inspection. C3 Rcos(x-a) max and min
A Assignment zeta Cover Sheet Name: Question Done Backpack Ready? Topic Comment Drill Consolidation M1 Prac Ch all Aa Ab Ac Ad Ae Af Ag Ah Ba C3 Modulus function Bb C3 Modulus function Bc C3 Modulus function
More informationI.I Stability Conditions
I.I Stability Conditions The conditions derived in section I.G are similar to the well known requirements for mechanical stability. A particle moving in an eternal potential U settles to a stable equilibrium
More information2.2 Relation Between Mathematical & Engineering Constants Isotropic Materials Orthotropic Materials
Chapter : lastic Constitutive quations of a Laminate.0 Introduction quations of Motion Symmetric of Stresses Tensorial and ngineering Strains Symmetry of Constitutive quations. Three-Dimensional Constitutive
More informationX 2 3. Derive state transition matrix and its properties [10M] 4. (a) Derive a state space representation of the following system [5M] 1
QUESTION BANK 6 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 5758 QUESTION BANK (DESCRIPTIVE) Subject with Code :SYSTEM THEORY(6EE75) Year &Sem: I-M.Tech& I-Sem UNIT-I
More informationc. Better work with components of slowness vector s (or wave + k z 2 = k 2 = (ωs) 2 = ω 2 /c 2. k=(kx,k z )
.50 Introduction to seismology /3/05 sophie michelet Today s class: ) Eikonal equation (basis of ray theory) ) Boundary conditions (Stein s book.3.0) 3) Snell s law Some remarks on what we discussed last
More informationAlgebraic Curves. (Com S 477/577 Notes) Yan-Bin Jia. Oct 17, 2017
Algebraic Curves (Com S 477/577 Notes) Yan-Bin Jia Oct 17, 2017 An algebraic curve is a curve which is described by a polynomial equation: f(x,y) = a ij x i y j = 0 in x and y. The degree of the curve
More informationThis theorem guarantees solutions to many problems you will encounter. exists, then f ( c)
Maimum and Minimum Values Etreme Value Theorem If f () is continuous on the closed interval [a, b], then f () achieves both a global (absolute) maimum and global minimum at some numbers c and d in [a,
More information10.4: WORKING WITH TAYLOR SERIES
04: WORKING WITH TAYLOR SERIES Contributed by OpenSta Mathematics at OpenSta CNX In the preceding section we defined Taylor series and showed how to find the Taylor series for several common functions
More informationOptimization Methods: Optimization using Calculus - Equality constraints 1. Module 2 Lecture Notes 4
Optimization Methods: Optimization using Calculus - Equality constraints Module Lecture Notes 4 Optimization of Functions of Multiple Variables subect to Equality Constraints Introduction In the previous
More informationUnconstrained Multivariate Optimization
Unconstrained Multivariate Optimization Multivariate optimization means optimization of a scalar function of a several variables: and has the general form: y = () min ( ) where () is a nonlinear scalar-valued
More informationCorner. Corners are the intersections of two edges of sufficiently different orientations.
2D Image Features Two dimensional image features are interesting local structures. They include junctions of different types like Y, T, X, and L. Much of the work on 2D features focuses on junction L,
More informationPhysics 828 Problem Set 7 Due Wednesday 02/24/2010
Physics 88 Problem Set 7 Due Wednesday /4/ 7)a)Consider the proton to be a uniformly charged sphere of radius f m Determine the correction to the s ground state energy 4 points) This is a standard problem
More informationThe Lanczos and conjugate gradient algorithms
The Lanczos and conjugate gradient algorithms Gérard MEURANT October, 2008 1 The Lanczos algorithm 2 The Lanczos algorithm in finite precision 3 The nonsymmetric Lanczos algorithm 4 The Golub Kahan bidiagonalization
More informationLight-induced spiral mass transport in azo-polymer films under vortex-beam illumination Supplementary Information
Light-induced spiral mass transport in azo-polymer films under vorte-beam illumination Supplementary Information Antonio Ambrosio a),1, Lorenzo Marrucci 1, Fabio Borbone, Antonio Roviello and Pasqualino
More informationTopic 8c Multi Variable Optimization
Course Instructor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) 747 6958 E Mail: rcrumpf@utep.edu Topic 8c Multi Variable Optimization EE 4386/5301 Computational Methods in EE Outline Mathematical Preliminaries
More informationvand v 3. Find the area of a parallelogram that has the given vectors as adjacent sides.
Name: Date: 1. Given the vectors u and v, find u vand v v. u= 8,6,2, v = 6, 3, 4 u v v v 2. Given the vectors u nd v, find the cross product and determine whether it is orthogonal to both u and v. u= 1,8,
More informationLattice dynamics. Javier Junquera. Philippe Ghosez. Andrei Postnikov
Lattice dynamics Javier Junquera Philippe Ghosez Andrei Postnikov Warm up: a little bit of notation Greek characters ( ) refer to atoms within the unit cell Latin characters ( ) refer to the different
More informationWeek Quadratic forms. Principal axes theorem. Text reference: this material corresponds to parts of sections 5.5, 8.2,
Math 051 W008 Margo Kondratieva Week 10-11 Quadratic forms Principal axes theorem Text reference: this material corresponds to parts of sections 55, 8, 83 89 Section 41 Motivation and introduction Consider
More informationEffective theory of quadratic degeneracies
Effective theory of quadratic degeneracies Y. D. Chong,* Xiao-Gang Wen, and Marin Soljačić Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received 28
More informationToday in Physics 217: begin electrostatics
Today in Physics 217: begin electrostatics Fields and potentials, and the Helmholtz theorem The empirical basis of electrostatics Coulomb s Law At right: the classic hand-to-thevan-de-graaf experiment.
More informationCHAPTER 8 The Quantum Theory of Motion
I. Translational motion. CHAPTER 8 The Quantum Theory of Motion A. Single particle in free space, 1-D. 1. Schrodinger eqn H ψ = Eψ! 2 2m d 2 dx 2 ψ = Eψ ; no boundary conditions 2. General solution: ψ
More information20 The Hydrogen Atom. Ze2 r R (20.1) H( r, R) = h2 2m 2 r h2 2M 2 R
20 The Hydrogen Atom 1. We want to solve the time independent Schrödinger Equation for the hydrogen atom. 2. There are two particles in the system, an electron and a nucleus, and so we can write the Hamiltonian
More informationMathematics. Mathematics 2. hsn.uk.net. Higher HSN22000
Higher Mathematics UNIT Mathematics HSN000 This document was produced speciall for the HSN.uk.net website, and we require that an copies or derivative works attribute the work to Higher Still Notes. For
More information