Patent Application: US-99436097-A

Abstract:
homotopy principles are used in computer simulation of magnetic resonance spectra . multidimensional homotopy provides an efficient method for accurately tracing energy levels , and hence transitions , in the presence of energy level anticrossings and looping transitions . the application describes the implementation of homotopy to the analysis of continuous wave electron paramagnetic resonance spectra . the method can also be applied to electron nuclear double resonance , electron spin echo envelope modulation , solid state nuclear magnetic resonance and nuclear quadrupole resonance spectra .

Description:
the basis of homotopy is the construction of smooth curves which connect the eigenpairs of one matrix to another . in the context of epr simulations , b res is assumed to be known ( calculated using matrix diagonalization , perturbation theory or any other method ) for some starting orientation . the implementation of homotopy to the analysis of randomly orientated epr spectra is described below . ______________________________________set up hamiltonian matrix defined by eq . [ 1 ] and [ 2 ] ( step1 ) calculate the eigenvectors and eigenvalues at θ . sub . 0 , φ . sub . 0bysolving hψ = eψlocate resonant field positions b . sub . res ( step 2 ) trace eigenpath from θ . sub . 0 , φ . sub . 0 to θ . sub . 1 , φ . sub . 1 ( step 3 ) set the tolerance δ , the maximum iteration number , nδθ and δφθ = θ . sub . 0 + δθφ = φ . sub . 0 + δφdouse e ( θ . sub . 0 , φ . sub . 0 , b . sub . 0 ) and e &# 39 ;( θ . sub . 0 , φ . sub . 0 , b . sub . 0 ) to predict e ( θ , φ , b . sub . 0 ) call rayleigh quotient iteration ( h , e . sub . i ( θ , φ , b . sub . 0 ), ψ . sub . i ( θ , φ , b . sub . 0 ), n , δ ) call rayleigh quotient iteration ( h , e . sub . j ( θ , φ , b . sub . 0 ), ψ . sub . j ( θ , φ , b . sub . 0 ), n , δ ) if rayleigh quotient iteration convergedθ . sub . 1 = θφ . sub . 1 = φexitelseδθ = δθ / 2θ = θ . sub . 0 + δθδφ = δφ / 2φ = φ + δφend ifwhile rayleigh quotient has not convergedendfind the resonant field position b . sub . res ( step 4 ) set the tolerance δ and the maximum iteration number , n . b = b . sub . 0dof . sub . b = e . sub . i ( θ . sub . 1 , φ . sub . 1 , b ) - e . sub . i ( θ . sub . 1 , φ . sub . 1 , b ) f . sub . b &# 39 ; = e . sub . i &# 39 ;( θ . sub . 1 , φ . sub . 1 , b ) - e . sub . j &# 39 ;( θ . sub . 1 , φ . sub . 1 , b ) δb = - f . sub . b / f . sub . b &# 39 ; b = b + δbcall rayleigh quotient iteration ( h , e . sub . i ( θ . sub . 1 , φ . sub . 1 , b ), ψ . sub . i ( θ . sub . 1 , φ . sub . 1 , b . sub . 0 ), n , δ ) call rayleigh quotient iteration ( h , e . sub . j ( θ . sub . 1 , φ . sub . 1 , b ), ψ . sub . j ( θ . sub . 1 , φ . sub . 1 , b ), n , δ ) if rayleigh quotient iteration convergedb . sub . res = bexitelseδθ = δθ / 2θ = θ . sub . 0 + δθδφ = δφ / 2φ = φ . sub . 0 + δφgo to step 3end ifend doend______________________________________ the rayleigh quotient iteration method [ 33 ], is used to determine both the eigenvalue and eigenvector of a symmetric matrix . ______________________________________rayleigh quotient iteration ( h , e , ψ , n , δ ) ψ . sub . 0 = ψ || ψ || μ . sub . 0 = efor i = 0 , ..., n - 1y . sub . i = ( h - μ . sub . i i ). sup .- 1 ψ . sub . iψ . sub . i + 1 = y . sub . i / || y . sub . i |. vertlineμ . sub . i + 1 = ψ . sub . i + 1 . sup . t hψ . sub . i + 1if (|| μ . sub . i + 1 - μ || & lt ; δ ) then = μ . sub . i + 1ψ = ψ . sub . i + 1return successend ifend for loopreturn failureend______________________________________ application of homotopy to a monotonically varying transition surface ( for example fig1 ) is fairly straightforward . given a single point on the surface , a method could just produce a line of eigenpair points along the θ axis , and then from every point on the line sweep out along the φ axis . however , such an algorithm will not find the complete surface if looping transitions are present ( fig2 ). using this fundamental method , there are three possible cases when the surface will not completed . the first two cases involve the comparison between two adjacent lines ( i . e . two lines along the φ axis ). if one line turns and the other doesn &# 39 ; t , or if there is a great difference in b values between adjacent points , then part of the surface could be missing , for example fig3 b . the third case is a surface that has a fairly complicated boundary and does not exist for the entire spatial dimension field . the following modifications to homotopy described herein provides a solution to these problems . in step three , homotopy attempts to trace the eigenvalues from one angular position to another . if homotopy is unsuccessful , then the spatial distance is halved and the step is repeated . if the spatial distance falls below a certain level , then there are two possibilities -- either the edge of the surface has been reached or a turn has been found . to investigate the possibility of a turn , homotopy is used to trace the eigenvalues in the reverse spatial direction . there is then the problem of whether a turn is being followed , or whether the previously found eignepairs are being rediscovered . to ensure that a turn is indeed being followed , the adjustments of b are checked . in general when b is adjusted , δb is confined to be within plus or minus a given tolerance , otherwise an error is produced . in the vicinity of a possible turn , δb is restricted further . if b was increasing prior to the possible turn , then b is confined to increase after then turn . if b was decreasing before the turn , then it will decrease after the turn . if homotopy continues to find eigenvalues with these restrictions imposed then a turn occurred , otherwise the edge of the surface was found . a comparison of the complexity analyses for matrix diagonalization (˜ 1000 n 3 m 2 ) and the homotopy - based method of this invention (˜ 4 / 3 d n 3 m 2 ), in table 1 , shows that providing the number of transitions ( d ) is less than 75 , the homotopy - based method will be computationally more efficient than matrix diagonalization . this method was tested on a high spin fe ( iii ) system s = 5 / 2 ( d = 0 . 1 cm - 1 , e / d = 0 . 25 , g = 2 . 0 ) for which the second order fine structure spin hamiltonian is with the microwave quantum ( ν = 9 . 0 ghz ) set to be slightly smaller than the zero - field splittings , multiple transitions can occur between a given pair of energy levels . for example , between levels 2 and 4 ( numbered in increasing energy ), three transitions at b res = 3 . 525 , 189 . 2 and 235 . 50 mt are detected . the power of homotopy in comparison to matrix diagonalisation is clearly demonstrated in fig1 , and 3 which compare transition surfaces from the two methods for particular transitions arising from an s = 5 / 2 spin system . fig1 shows the transition surfaces for levels 5 and 6 . the homotopy surface ( fig1 a ) shows exactly the same structure as calculated by matrix diagonalization ( fig1 b ). this transition surface varies monotonically as a function of the euler angles and the magnetic field b . table 2 gives the results of comparison between homotopy and the matrix diagonalization , where the following information is given in the columns : 2 . the relative difference ( ra ) between the sophe points calculated using the two methods , ## equ3 ## 3 . the total number of sophe points computed , 4 . the number of erroneous data points where the magnetic filed value for matrix diagonalization and homotopy do not agree , or where matrix diagonalizsation found an erroneous b value and 5 . the number of multiple valued sophe points calculated by homotopy for which the matrix diagonalization routine may have calculated multiple points by sophe was unable to complete the surface . consequently only the lowest valued point has been reported . clearly , homotopy reproduces the surface obtained by matrix diagonalization between levels 5 and 6 as the relative error between the two methods is very small ( 4 . 104334 × 10 - 5 ). the transition surface from level 3 to level 5 is a looping transition as shown in fig2 a , which is produced by homotopy , but not by matrix diagonalization ( fig2 b ). in fact , matrix diagonalization calculates an erroneous surface since no transitions between levels 3 and 5 exist when θ & gt ; 40 °. in fig3 an anti - level crossing between levels 2 and 4 is graphed . homotopy ( fig3 a ) resolves the complete structure while matrix diagonalization completes only the mostly monotonic lower transition ( fig3 b ). in this case matrix diagonalization may have found multiple points . matrix diagonalization may have discarded some points in completing the surface , as it is unable to make a connection between the multivalued points . note that the unmatched homotopy points indicate structure not revealed with matrix diagonalization , while the points where matrix diagonalization and homotopy do not agree indicate erroneous points calculated by matrix diagonalization . the homotopy method is not only capable of obtaining b res but also automatically establishes the connection for b res between the grid points . this will allow the highly efficient sophe interpolation scheme to be used to interpolate b res at any point between the grid points . in addition , homotopy does a search in frequency space for the resonant frequency . during the search for the resonant frequency , the eigenvectors and eigenvalues are calculated at various points across a given resonance . while the eigenvalues are required to calculate the line shape , the eigenvectors may be used to calculate the transition probability , which is known in some spin systems to vary across resonances . thus , homotopy allows computer simulation of spectra in frequency space , which is the correct way of simulating spectra . since homotopy traces the eigenpairs it will automatically solve the energy level anti - crossing problem and the looping transition problem . in conjunction with sophe the homotopy method leads to improved quality of simulated spectra , allows the analysis of more complicated epr spectra and reduces the computational time . although the invention has been described with particular reference to computer simulation of cw - epr spectra , the method of this invention can be used , with appropriate modification where necessary , for the simulation of field dependent endor , eseem , pulsed endor , solid state nmr and nuclear quadrupole resonance spectra . table 1______________________________________procedure computation time______________________________________matrix diagonalizationtri - diagonalize h matrix . 2 / 3 n . sup . 3diagonalize h matrix with an iterative qr method . sup . 1 8 - 10 n . sup . 3repeat diagonalization 200 times for each ˜ 2000 n . sup . 3orientation ( θ , φ ). total computation time for m . sup . 2 / 2 orientations . ˜ 2000 n . sup . 3 m . sup . 2 / 2homotopytri - dagonalize h matrix . 2 / 3 n . sup . 3rayleigh quotient iteration . sup . 2 15 - 20 nfind ψ from the original h matrix . 2 n . sup . 2repeat ˜ 4 times to trace transition surface from ˜ 8 / 3 n . sup . 3one orientation to another . this calculation is repeated for d allowed ˜ 8 / 3 d n . sup . 3transitions . total computation time for m . sup . 2 / 2 angular ˜ 8 / 3 d n . sup . 3 m . sup . 2 / 2orientations . ______________________________________ . sup . 1 determine all e , ψ of tridiagonal h matrix . . sup . 2 using the tridiagonal h matrix . table 2______________________________________ total erroneous unmatched relative sophe data homotopylevels accuracy ( ra ) points ( n ) points . sup . 1 points . sup . 2______________________________________2 4 1 . 071440 × 10 . sup .- 3 666 2 1563 5 1 . 025706 × 10 . sup .- 4 666 411 2555 6 4 . 104334 × 10 . sup .- 5 666 0 0______________________________________ . sup . 1 the number of erroneous data points where the magnetic field value for matrix diagonalization and homotopy do not agree , or where matrix diagonalization found an erroneous b value . . sup . 2 the number of multiple valued sophe points calculated by homotopy for which matrix diagonalization did not return a multiple valued point . a . abragam and b . bleaney . electron paramagnetic resonance of transition ions . clarendon press , oxford , 1970 . f . e . mabbs and d . c . collison . electron paramagnetic resonance of transition metal compounds . elsevier , amsterdam , 1992 . j . r . pilbrow . transition ion electron paramagnetic resonance . clarendon press , oxford , 1990 . a . bencini and d . gatteschi . epr of exchange coupled systems . springer - verlag , berlin , 1990 . t . d . smith and j . r . pilbrow . coord . chem . rev ., 13 : 173 , 1974 . 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