Some Practical Suggestions for Optimizing Geometries and Locating
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1. Gaw J F Yamaguchi Y Remington R B Osamura Y and Schaefer H F Chem Phys 109 237 1986 Schaefer H F and Yamaguchi Y J Mol Struct 135 369 1986 Gaw J F and Handy N C in 6 Duran M Yamaguchi Y Osamura Y and Schaefer H F J Mol Struct 163 389 1988 Pulay P J Chem Phys 78 5043 1983 Alml f J and Taylor P R Int J Quantum Chem 27 743 1985 J rgensen P and Simons J J Chem Phys 79 334 1983 Schlegel H B in Computational Theoretical Organic Chemistry Eds Csizmadia I G and Daudel R Reidel Dordrecht 1981 Schlegel H B Adv Chem Phys 67 249 1987 Bell S and Crighton J S J Chem Phys 80 2464 1984 Head J D Weiner B and Zerner M C Int J Quantum Chem 33 177 1988 Ss L E Introduction to Non linear Optimization MacMillan Basingstoke 1985 Fletcher R Practical Methods of Optimization Wiley Chichester 1981 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 46 47 48 49 50 51 52 53 54 55 56 ST 53 Gill P E Murray W and Wright M H Practical Optimization Academic Press New York 1982 Powell R Non linear Optimization Academic Press New York 1982 Clark T A Handbook of Computational Chemistry Wiley Interscience New York 1985 Stanton R E and Mclver Jr J W J Am Chem Soc2. and the distance 1 48 A are required The third line indicates that atom 3 is an H bonded to atom 2 ata distance of 0 96 A and makes an angle with atom 1 of 105 0 i e lt 321 Atom 4 is an H bonded to atom 1 at a distance of 1 08 A makes a valence angle with 2 of 109 5 lt 412 and a dihedral angle with 3 of 180 i e lt 4123 the angle of rotation between 4 and 3 about the axis formed by atoms 1 and 2 Subsequent atoms are defined in a similar way in terms of a distance valence angle and a dihedral angle to previously defined atoms In some programs e g GAUSSIAN a second valence angle can be used instead of the dihedral angle see Example 7 The appropriate user manual should be consulted for the details of specifying internal coordinates for a particular MO program 37 Example 1 Z matrix for methanol C H4 O 1 1 48 H 2 0 96 1 105 0 H 1 1 08 2 109 5 3 180 Gira C2 H 1 1 08 2 109 5 4 120 He H 1 1 08 2 109 5 4 120 Hs H3 In addition to real atoms most programs allow dummy atoms to be used in the description of the geometry Dummy atoms have no charge and carry no basis functions they are used solely to help define the geometry so that a more convenient set of distances and angles can be used Dummy atoms are identified by a particular atomic symbol or atomic number X or 1 in GAUSSIAN see Example 3 and are specified in the same manner as regular atoms Some examples that are discussed below include linea
3. 433 1985 Adamowicz L Laidig W D and Bartlett R J Int J Quantum Chem Symp 18 245 1984 Kato S and Morokuma K Chem Phys Lett 65 19 1979 Goddard J D Handy N C and Schaefer H F J Chem Phys 71 1525 1979 Schlegel H B and Robb M A Chem Phys Lett 92 43 1982 Knowles P J Sexton G J and Handy N C Chem Phys 72 337 1982 Taylor P R J Comput Chem 5 589 1984 Pople J A Krishnan R Schlegei H B and Binkley J S Int J Quantum Chem Symp 13 225 1979 Handy N C Amos R D Gaw J F Rice J E Sirnandiras T J Lee T J Harrison R J Laidig W D Fitzgerald G and Bartlett R J in ref 6 Handy N C Amos R D Gaw J F Rice J E and Simandiras T J Chem Phys Lett 120 151 1985 Lee T J Handy N C Rice J E Scheiner A C and Schaefer H F J Chem Phys 85 3930 1986 Camp R N King H F McIver J W and Mullally D J Chem Phys 79 1088 1983 Hoffman M R Fox D F Gaw J F Osamura Y Yamaguchi Y Grev R S Fitzgerald G Schaefer H F Knowles P J and Handy N C J Chem Phys 80 2660 1984 Page M Saxe P Adams G F and Lengsfield J Chem Phys 81 434 1984 Gaw J F Yamaguchi Y and Schaefer H F J Chem Phys 81 6395 1984 Gaw J F Yamaguchi Y Schaefer H F and Handy N C J Chem Phys 85 5132 1986
4. 97 3632 1975 Thiel W J Mol Struct 163 415 1988 a The author wishes to thank Dr M J Frisch for the guidelines for using symmetry to place dummy atoms These algorithms have been incorporated in the program NewZmat 33b b Frisch M J NewZmat 1986 Pople J A J Am Chem Soc 102 4615 1980 Pople J A Sataty Y A and Halevi E A Israel J Chem 19 290 1980 Pople J A and Gordon M S J Am Chem Soc 89 4253 1967 Radom L Hehre W J and Pople J A J Am Chem Soc 93 289 1971 Peterson M R and Csizmadia 1 G J Mol Struct 125 399 1985 Burkert U and Allinger N L Molecular Mechanics American Chemical Society 1981 Sadlej J Cooper I L transl ed Semi empirical Methods of Quantum Chemistry Ellis Horwood Chichester 1985 Segal G A Semi empirical Methods of Electronic Structure Calculation Modern Theoretical Chemistry Vol 7 and 8 Plenum New York 1977 Halgren T A and Lipscomb W N Chem Phys Lett 49 225 1977 Cerjan C J and Miller W H J Chem Phys 75 2800 1981 Simons J J rgensen P Taylor H and Ozment J J Chem Phys 87 2745 1983 Nguyen D T and Case D A J Phys Chem 89 4020 1985 Banerjee A Adams N Simons J and Shepard J Phys Chem 89 52 1985 Hoffman D K Nord R S Ruedenberg K Theor Chim Acta 69 265 1986 J rgensen P Jensen H J
5. A and Helgaker T Theor Chim Acta 73 55 1988 Baker J J Comput Chem 7 385 1986 Schlegel H B J Comput Chem 3 214 1982 Scharfenb rger P J Comput Chem 3 277 1982 Tapia O and Andr s J Chem Phys Lett 109 471 1984 Bell S Crighton J S and Fletcher R Chem Phys Lett 82 122 1981 A J D Weiner B and Zemer M C Int J Quantum Chem 33 177 700 B lint I and B n M I Theor Chim Acta 63 255 1983 Schlegel H B Theor Chim Acta 66 333 1984 Under some circumstances an imaginary frequency may also signal an instability in the wavefunction Ishida K Morokuma K and Komornicki A J Chem Phys 66 2153 1977 Schmidt M W Gordon M S and Dupuis M J Am Chem Soc 107 2585 1985 M ller K and Brown L D Theor Chim Acta 53 75 1979 Garrett B C Redmon M J Steckler R Truhlar D G Baldridge K K Bartol D Schmidt M W and Gordon M S J Phys Chem 92 1476 1988 Page M and McIver Jr J W J Chem Phys 88 922 1988 Gonzalez C Schlegel H B submitted also described in ref 23 Fletcher R and Powell M J D Comput J 6 163 1963 Davidon W ioe Nat Lab Report ANL 5990 Binkley J S J Chem Phys 64 5142 Murtagh B A and Sargent R W H Comput J 13 185 1972
6. for any of the other angles Scheme 1 2 cos a sina 0 2 3 cos 7 cos y cos acos sin a sin B cos T 1 0 0 cos 6 cos y cos a cos sin a sin B 3 cos 8 sin B cos sin B sin 8 47 Scheme 2 deals with the more specialized case when a plane or axis bisects an angle The analysis proceeds as before but with the bisector placed on the axis If the planes of the angles a and B are perpendicular a particularly simple expression is obtained cos Y cos o 2 cos B Many other relations can be derived in the same manner The key is to place the right vector on the axis so that the remaining vectors can be generated by simple rotations about axes Scheme 2 2 cos o 2 sin w2 0 2 3 cos Y cos Y cos o 2 cos Bf sin o 2 sin B cos 2 cos o 2 sin o 2 0 3 cos B sin B cos 8 sin B sin 8 11 Estimating the Hessian Like the choice of internal coordinate systems and the starting geometry the initial estimate of the Hessian can strongly influence the rate of convergence of an optimization As indicated in eq 3 the next estimate of the optimized geometry depends on the Hessian The better the initial Hessian the better the predicted geometry and fewer steps needed to converge to the optimized geometry the final optimized geometry of course is independent of the Hessian At each step in the optimization the approximate Hessian is updated so that it eventually approaches the corre
7. other calculations are referenced in Quantum Chemical Literature Data Base 2 also available on line The molecule may have already been optimized at the desired level of theory or at a lower level Alternatively these data bases are an excellent source for fragment geometries from which the molecule in question can be constructed f Experiment X ray microwave and electron diffraction structures are highly desirable starting points for geometry optimizations but frequently are not available for the structures studied theoretically 9 Getting Close to the Transition Structure Often a transition structure can be optimized directly without any problems especially if the initial estimate of the Hessian has a suitable negative eigenvalue with an eigenvector that is a reasonable estimate to the transition vector see Section 11 However convergence to a transition state may sometimes require a better starting geometry than an optimization of aminimum For difficult cases standard geometries and chemical intuition may not yield estimates that are good enough for a direct transition structure optimization Various techniques for getting close to a transition structure have been discussed in the literature for leading references see 23 A few of the methods are summarized below in order of increasing sophistication a Potential surface scan If there is some doubt about the location of a saddle point along a reaction path a series of en
8. proceeding to a higher symmetry structure If a higher symmetry structure is suspected it is often more efficient to optimize the higher symmetry structure directly and test it for stability with respect to distortion to lower symmetry Section 12 Some transition structure optimizations can be turned into minimizations by symmetry Stanton and McIver 30 have discussed a number of symmetry restrictions on transitions states the ones directly relevant to transition structure optimization can be summarized as follows a a transition vector cannot belong to a degenerate representation otherwise there would be at least 2 equivalent eigenvectors of the Hessian with negative eigenvalues and the structure is at least a second order saddle rather than a first order saddle point b the transition vector must be symmetric for all symmetry operations of the transition state that also leave the reactants and products unchanged c the transition vector must be antisymmetric for any of the symmetry operations of the transition state that interconvert the reactants and products For most reactions only a and b can be used to simplify the transition structure optimization However in some reactions such as racemizations degenerate isomerizations identity exchanges etc reactants and products can be interchanged by a symmetry operation of the transition state e g a C2 axis for the 1 3 antara hydrogen shift CH3CH CH2 CH2 CHCH3 a o pla
9. 3ACL 2BCL se H 4RHCL 1 AHCL 3 BHCL F3 Strong coupling between coordinates can be particularly troublesome in transition structure optimizations As far as possible the transition vector should be dominated by only a few coordinates preferably 1 or 2 The coupling between the coordinates of the transition vector and the remaining coordinates should also be as small as possible Two coordinate systems for HCN isomerization are shown in Example 11 In the first case the transition vector involves both the HCN bend and the CH stretch but in the second case the transition vector is predominantly the XH displacement _ Example 11 Symbolic Z matrices for the HCN HNC transition state a simple coordinates b better coordinates C C N 1 RCN N 1 RCN H 1 RCH 2 AHCH X 1 RCX 2 90 H 3 RXH 1 90 20 H3 X3 H4 l 1 1 I C N2 Cy N2 For some reactions the coordinate system is easy to set up For Sn2 reactions abstractions and one center addition reactions the transition vector is dominated by the bonds being made or broken For other reactions such as insertions eliminations and cycloadditions the choice of coordinates is less clear These combine the difficulties of cyclic structures and loose complexes with the problems of transition states In the elimination of H from H2CO Example 12 the hydrogens could be specified by distances to the carbon and angles to CO however a small rotation of Hz would require t
10. Cartesian coordinates are available it is straight forward but tedious to determine the new internal coordinates manually Alternatively there are some programs that construct a new Z matrix automatically 32b Most general purpose molecular modelling programs can also be used to calculate bond lengths and angles from the Cartesian coordinates However there may be some circumstances in which it is necessary or more convenient to interconvert a few valence and dihedral angles manually Schemes 1 and 2 show the construction of 2 of the more frequently used relations between valence and dihedral angles In both cases the Cartesian coordinates of appropriate fragments are constructed by rotating unit vectors by the required angles about particular axes The desired angles are then obtained by simple dot products Scheme 1 deals with the angles about a non planar tricoordinate center or a tetracoordinate center given valence angles a B and y find dihedral angle 6 or given o B and find v Vector 1 is placed on the x axis 1 0 0 vector 2 is obtained by rotating vector 1 by zbout the z axis i e in the x y plane to give cos a sin a 0 Vector 3 is obtained by rotating vector 1 by B about the z axis cos B sin B 0 followed by amp about the x axis to give cos p sin B cos 6 sin B sin 8 The dot product between vectors 2 and 3 is the cosine of the third valence angle y If needed the resulting expression for cos y can be solved
11. Some Practical Suggestions for Optimizing Geometries and Locating Transition States H Bernhard Schlegel Department of Chemistry Wayne State University Detroit Michigan 48202 USA ABSTRACT The optimization of equilibrium geometries and transition states by molecular orbital methods is discussed from a practical point of view Most of the efficient geometry optimization methods rely on analytical energy gradients and quasi Newton algorithms For any optimization method there are three areas of input that directly affect the behavior of the optimization a the choice of internal coordinates b the starting geometry and c the initial estimate of the Hessian A number of topics related to these three areas are discussed with the aim of improving the performance of optimizations these include symmetry dummy atoms avoiding coordinate redundancy overcoming strong coupling among coordinates conversion between coordinate systems testing stationary points and what to do when optimizations fail 1 Introduction Equilibrium geometries can be calculated routinely reliably and accurately by ab initio molecular orbital methods 1 In principle transition states can be calculated equally well though in practice transition structures require a little more skill to optimize than equilibrium structures The importance of geometry optimization is supported by the fact that most molecular orbital calculations appearing in the chemical literature
12. This approach can be quite costly if the method requires the Hessian to be recomputed frequently The idea in both methods is to construct a path of shallowest ascent toward the transition structure 46 However both methods can miss the transition state if it is not on the shallowest ascent path see 23 for examples Once a reasonable initial estimate has been obtained for the transition structure a suitable gradient optimization method 43 48 can be used to find the optimized geometry If the initial estimate has been obtained by a potential surface scan or a linear synchronous transit calculation it may be preferable to optimize the transition structure in 2 or more phases First the coordinates corresponding to the transition vector are frozen and the remaining coordinates are minimized This brings the molecule closer to the reaction path The second step is a full transition structure optimization with all coordinates varied Some of the hill climbing and eigenvector following methods incorporate the equivalent of a direct gradient optimization as a final step 10 Converting Between Different Z Matrices In setting up a calculation or during the course of an optimization it is sometimes necessary to convert from one set of internal coordinates to another e g the experimental geometry may be specified in an inconvenient coordinate system or an attempt at optimization may reveal a strong coupling between coordinates If the
13. action displace along this eigenvector until a minimum is found and restart the transition structure optimization from the new geometry with a new estimate of the Hessian e g re compute the Hessian elements for the displaced coordinates The remedy for case b is the same as above restart with a better estimate of the Hessian Like with minimizations if the problems persist it may be necessary to freeze a few of the more flexible coordinates until the optimization gets closer to the saddle point No negative eigenvalues of the Hessian during a transition structure optimization There are 2 possible causes a the structure is not a saddle point or is not sufficiently close to the quadratic region of the saddle point or b there are numerical problems with the Hessian For a any of the methods for getting closer to the transition state can be tried Section 9 i linear synchronous transit starting from the current structure plus a corresponding structure on the other side of the transition state ii coordinate driving several steps of increment the coordinate dominating the reaction path and minimizing of the rest of the coordinates or iii eigenvector following choose the appropriate eigenvector and use one of the hill climbing algorithms to follow the path of shallowest ascent toward the transition structure Once a more suitable starting structure has been found one of the direct methods for transition structure optimization c
14. an be used The remedy for case b is the same as above restart with a better estimate of the Hessian 50 Eigenvalue of the Hessian too large While this is not normally a problem it may signal an error in the input This may be due to a a bad initial Hessian get a better estimate b a bad update of the Hessian restart with a new estimate of the Hessian or c a strongly coupled coordinate system reconstruct the Z matrix to avoid the strong coupling Ejigenval Hessian mall There are a number of possible causes for this problem a the molecule is actually in a shallow minimum and should have small eigenvalues b there is a redundancy in the choice of internal coordinates or c there are numerical problems with the Hessian For a it may be necessary to tighten the convergence criteria e g RMS gradient as well as the optimization control parameters like the test for small eigenvalues Case b is the more likely problem and requires that the Z matrix be reconstructed to remove the redundancy see Section 6 The remedy for c is to restart with a better estimate of the Hessian Number of steps exceeded The possibilities are a it really is a difficult optimization and needs more steps b there is a redundancy in the internal coordinates fix the Z matrix and restart c there are some very loose coordinates that are slowing the optimization freeze the loose coordinates until the others have converged then
15. ari K Schlegel H B Whiteside R A Fox D J Martin R L Fluder E M Melius C F Kahn L R Stewart J J P Bobrowicz F W and Pople J A GAUSSIAN 86 Carnegie Mellon Publishing Unit Pittsburgh 1984 and subsequent releases 5 Pulay P Adv Chem Phys 69 241 1987 6 J rgensen P and Simons J Eds Geometrical Derivatives of Energy Surfaces and Molecular Properties Reidel Dordrecht 1986 7 T F and Handy N C Annu Rep Prog Chem Sec C 81 291 8 Fogarasi G and Pulay P Annu Rev Phys Chem 35 191 1984 i5 16 17 18 19 20 21 22 23 24 25 26 27 Fitzgerald G Harrison R Laidig W D and Bartlett R J J Chem Phys 82 4379 1985 Gauss J and Cremer D Chem Phys Lett 138 131 1987 Krishnan R Schlegel H B and Pople J A J Chem Phys 72 4654 1980 Brooks B R Laidig W D Saxe P Goddard J D Yamaguchi Y and Schaefer H F J Chem Phys 72 4652 1980 Osamura Y Yamaguchi Y and Schaefer H F J Chem Phys 77 383 1982 Rice J E Amos R D Handy N C Lee T J and Schaefer H F J Chem Phys 85 963 1986 Shepard R Int J Quantum Chem 31 33 1987 Scheiner A C Scuseria G E Rice J E Lee T J and Schaefer H F J Chem Phys 87 5361 1987 Fitzgerald G Harrison R Laidig W D and Bartlett R J Chem Phys Lett 117
16. behavior during the optimization If there are no dummy atoms and no symmetry the Z matrix input contains 3n 6 unique parameters all of which must be varied in a full optimization When dummy atoms are present some of the Z matrix parameters are redundant and must be held fixed In Example 5 a dummy atom is needed to specify a nearly linear NCO angle Clearly changing the distance to the dummy atom RCX does not alter the position of the real atoms Hence RCX is redundant and must not be included in the optimization A more subtle redundancy exists between the angles about the dummy atom ANCX and AXCO A change in either will bend the NCO angle But if one angle is increased and the other is decreased by the same amount the NCO angle remains the same Hence only one of the angles can be included in the optimization and the other must remain fixed as shown in Example 3 the 2 dihedral angles are also fixed because the molecule is planar Example 5 Symbolic Z matrix for HOCN with errors due to redundant coordinates C X3 N 1 RCN X 1 RCX 2 ANCX O 1 RCO 3 AXCO 2 180 No C 04 H 4 ROH 1 ACOH 3 180 Hs 40 If the molecule has some symmetry there are fewer than 3n 6 parameters to optimize if symmetry is to be retained during the optimization The number of parameters to be optimized can be determined from the symmetries of the normal modes of vibration of the molecule Only displacements along totally symmetric vibrationa
17. ble to construct a set of standard values of similar quality for the coordinates undergoing changes in transition states However concepts such as conservation of bond order bond order bond energy relations and the Pauling relation between bond length and bond order may be quite helpful b Empirical force field calculations Although the range of molecules and the types of bonds can be somewhat limited molecular mechanics calculations give good estimates of the optimized geometry for energy minima 36 especially of cyclic systems and cases where steric interactions are important However some care is needed with these methods in situations where electronic factors control the geometry c Semi empirical MO calculations Some semi empirical MO programs predict optimized geometries as well as minimal basis set ab initio calculations Known defects of various methods 29 37 38 should be taken into account when using semi empirically optimized geometries for ab initio calculations 45 d Lower level ab initio calculations The various shortcomings of smaller basis sets 1 should be taken into account when scaling to calculations with larger basis sets and or electron correlation e Quantum chemical data bases Quite a wide variety of molecules have been optimized over the past decade or two A large number of structures are available in machine readable and searchable form in the Carnegie Mellon Quantum Chemistry Archive 3 Many
18. ct Hessian Thus with a poor initial Hessian many of the optimization steps are needed just to improve the Hessian For a minimization the initial Hessian must at least be positive definite i e no zero or negative eigenvalues For a transition structure or first order saddle point a Hessian must have one and only one negative eigenvalue and the corresponding eigenvector must be a reasonable approximation to the transition vector Several alternatives are available for estimating the initial Hessian most are automated in the more flexible MO programs In order of increasing cost these include a Unit matrix or a unit matrix scaled by a constant This contains no useful structural data about the molecule Information about the stiffness or flexibility of various modes and the coupling between coordinates must be accumulated via the Hessian updating scheme during the course of the optimization This will increase the number of optimization steps substantially A scaled or unscaled unit is unsuitable for a saddle point optimization because it does not provide an estimate of the transition vector b Empirical force field Hessian Molecular mechanics force fields can be quite good for minima but the types of molecules and bonding situations treated by these force fields can be somewhat limited A less accurate but more general scheme for estimating the Hessian from a simple valence force field has been used successfully 48 for mi
19. ed to define an internal coordinate system For acyclic molecules this is quite easy in the Z matrix format The use of symmetry is straight forward and often dummy atoms are not needed However some care must be taken with torsional modes and with nearly planar trigonal centers Example 6 illustrates the problem of methyl rotation in CH3SHO Each of the hydrogens in the methyl group could be defined with a dihedral angle to the SO bond however this would require all 3 angles to change by the same amount if the methyl group is rotated in the course of an optimization Alternatively a single dihedral angle can be used to specify the rotation of the entire methyl group thereby decoupling the flexible mode methyl rotation from the stiff modes HCH bend The same idea can be applied to any group that can rotate Example 6 Symbolic Z matrix for methyl sulphoxide a Methyl rotation strongly coupled b Methyl rotation decoupled from HCH bend S S Hs H3 O 1 RSO O 1 RSO D H 1 RSH 2 AOSH H 1 RSH 2 AOSH C 1 RSC 2 AOSC 3 DHOSC C 1 RSC 2 AOSC 3 DHOSC C4 S H 4 RCH1 1 ACHI 2 BCHI H 4 RCH1 1 ACHI 2 BCHI H7 H 4 RCH2 ACH2 2 BCH2 H 4 RCH2 1 ACH2 5 DCH2 H 4 RCH3 1 ACH3 2 BCH3 H 4 RCH3 1 ACH3 5 DCH3 He O2 Example 7 shows pyramidal CH3 that will become planar during the optimization If 3 valence angles are used to define a nearly planar geometry then a small change in the angles will cause a large out of plane displacement However if a d
20. ents first a If there is an atom at the center of inversion the intersection of 2 or more rotational axes or the intersection of an axis and a perpendicular plane specify that atom first e g S in SFe If there is no atom at the center of symmetry put a dummy atom there e g the middle of the ring in cyclopropane C3H6 b If there are axes of rotation specify the atoms on the highest order axis first e g C and F on the C3 axis in CH3F If there is only 1 atom on the axis add a dummy atom on the axis e g in the direction of the lone pair NH3 or on the bisector of lt HCH and lt FCF in CH2F2 Some cases such as rings may require 2 dummy atoms to define the axis e g C3H6 c If there is only a plane of symmetry and no other elements of symmetry specify at least 2 and preferably 3 atoms in the plane first e g N O and H in H2NOH If there are less than 2 or 3 atoms in the plane add enough dummy atoms to define the plane so that the remaining atoms can be specified relative to the plane In specifying the other atoms in the molecule it is best to group them into symmetry equivalent sets e g the 3 C s in cyclopropane form one set and the 6 H s form another set For each atom in a symmetry equivalent set d Define the bond lengths and angles in symmetry equivalent ways e g in CH3F each H is bonded to C and makes an angle with F e Use the same variable name for equivalent bond lengths valence angles and di
21. ergy only or energy gradient calculations may be sufficient to locate the region of the transition state approximately If more than one coordinate is important in the reaction path a small grid of points may have to be calculated b Linear synchronous transit LST 39 In this approach the reaction path is approximated as a straight line in the space of interatomic distances this may correspond to a curved path in Cartesian coordinates or internal coordinates An estimate of the transition structure is obtained by finding the maximum along this one dimensional path Since the true reaction path usually differs from the LST path the LST estimate of the transition state is normally higher in energy than the true transition state and may be outside the quadratic region of the true transition state However if a minimization is carried out perpendicular to the reaction path akin to quadratic synchronous transit a better estimate of the transition state can be obtained c Coordinate driving walking up valleys and eigenvector following methods 40 42 These methods attempt to follow the reaction path uphill toward the transition structure In the coordinate driving approach a coordinate dominating the reaction path is incremented at each step and the remaining coordinates are minimized In the eigenvector following approach steps are taken in the uphill direction along a selected eigenvector usually the one with the lowest eigenvalue
22. ese steps in common thus almost all optimization methods need a starting geometry and an initial estimate of the Hessian Much of this Chapter will deal with the choice of internal coordinate system for an optimization the starting geometry and the initial estimate of the Hessian The various optimization algorithms differ in the way the estimated Hessian is updated at each step e g DFP MS BFGS etc how the solution of eq 3 is constrained e g reduction of steps that are larger than the trust radius and how the one dimensional optimization is carried out along the direction predicted by eq 3 e g accurate approximate or not at all The details can be found in a various books and review articles 23 28 Given the same starting conditions most of the recent gradient algorithms for minimizations perform similarly The BFGS method with a very approximate line search seems to be quite efficient for geometry minimization 31 Aside from GAUSSIAN see Appendix most molecular orbital programs have only a limited selection of optimization algorithms and one must work within the framework of the available code Thus success or failure in an optimization using a particular program will depend significantly on the starting conditions of the optimization The 3 most important areas are a choice of the internal coordinates Sections 4 7 b the starting geometry Sections 8 10 and c the initial estimate of the Hessian Section 11 4 C
23. exceptions to this practice If there is any doubt the stationary point should be tested at the highest level of calculation used for the optimization For transition structures it is also important to check the nature of the eigenvector with the negative eigenvalue to be sure that the saddle point connects the correct reactants and products For some reactions especially those involving non least motion pathways it may not be immediately obvious from the transition vector that the appropriate saddle point for the reaction has been found In such cases it may be necessary to follow the reaction path part of the way from the saddle point toward the reactants and toward the products to verify that the transition state is on the correct reaction path A number of 49 algorithms for reaction path following have been published 51 55 The most efficient method for following paths by MO calculations appears to be a recent method by Gonzalez and Schlegel 55 13 Things to Try When Optimizations Fail _ Optimizations misbehave for a variety of reasons Listed below are a number of conditions that can be encountered during a minimization or a search for a saddle point Some optimizers check for these problems reporting them when they occur and stopping if necessary Other optimizers struggle to continue and may terminate without warnings or diagnostics when the situation becomes hopeless For conditions listed below some possible causes are given a
24. he concerted change of 2 distances and 2 angles The alternate coordinate system avoids this problem and also reduces the coupling to the transition vector primarily CX and XH stretch Example 13 deals with the 1 2 cycloaddition of Hz to N2H2 In this case symmetry is used to construct a less coupled coordinate system by locating the dummy atoms on the C axis as discussed in Section 5 44 Example 12 Symbolic Z matrices for the H2CO E2 CO transition state a simple coordinates b better coordinates Cc Cc O 1 RCO O 1RCO H 1 RCH1 2 ACHI X 1 RCX 2 AOCX H 1 RCH2 2 ACH2 H 3 RXH 1 ACXH 20 X310 4 90 1 180 H 3 RXH 5 90 4 180 Xs Hg Pa He Hw g C O2 Cy O2 Example 13 Symbolic Z matrix for the H2 N2H2 N2H4 transition state x X 1 RXX H4 X2 H3 H 2 RXH 1 90 l H 2 RXH 1 90 3 180 9 N 1 RXN 2 90 3 0 N 1 RXN 2 90 3 180 Ns X N5 H 5 RNH 1 ANH 2 BNH H 6 RNH 1 ANH 2 BNH 2 Hg H7 8 Initial Guess for Bond Lengths and Angles One of the simplest ways to speed up an optimization is to make a good estimate of the geometry Initial values for bond lengths and angles can be obtained from a number of sources a Standard geometries There are various compilations of standard bond lengths and angles for a wide range of equilibrium structures 34 35 With a bit of chemical intuition and VSEPR theory these standard values can be adjusted to give better estimates It is not possi
25. hedral angles caution while valence angles do not change under any symmetry operation dihedral angles must change sign on reflection or improper rotation 39 Example 4 Symbolic Z matrix for cyclopropane illustrating the use of dummy atoms and symmetry x Hi0 X110 C 1 RXC 2 90 X2 He C 1 RXC 2 90 3 120 Ha C5 C 1 RXC 2 90 3 240 V Sx ce H 3 RCH 1 AXCH 2 0 S 3 H 3 RCH 1 AXCH 2 180 Hii cd H 4 RCH 1 AXCH 2 0 Hy H 4 RCH 1 AXCH 2 180 H 5 RCH 1 AXCH 2 0 He H 5 RCH 1 AXCH 2 180 RXC 0 8735 RCH 1 072 AXCH 122 6 6 Full Optimizations and the Number of Internal Coordinates Whenever possible a full optimization should be carried out With the current gradient based optimization methods most full optimizations are not appreciably longer than partial optimizations and one has the assurance that subtle features of the energy surface are not masked by artificial geometric constraints For a molecule without symmetry there are 3n 6 internal coordinates where n is the number of atoms and all 3n 6 degrees of freedom must be varied in a full optimization A frequent cause of difficulty in optimizations is the specification of too few or too many coordinates Too few coordinates corresponds to a constraint on the geometry of the molecule and results in a less than complete optimization Too many coordinates results in a redundancy among the internal coordinates leading to zero eigenvalues in the Hessian and to very poor
26. here is only a center of inversion or one symmetry plane Cj Cs e 6 if there is no symmetry C1 This procedure can be illustrated by examining pyramidal trimethylamine N CH3 3 which has C3 symmetry The N is on the C3 axis 1 3 equivalent C s are in the oy planes 2 3 equivalent H s are in the oy planes 2 6 equivalent H s are not on any symmetry element 3 and finally subtract 1 for rotation and translation because of the oy planes and a C3 axis Thus the total number of coordinates to be optimized is 14 2 2 3 1 7 7 Strong Coupling between Internal Coordinates Many of the problems encountered in optimizations see Section 13 can be traced to a poor choice of coordinate system In general strong coupling between coordinates degrades the performance of any optimization algorithm In particular coupling between stiff modes e g bond stretch angle bend etc and flexible modes e g torsion about single bonds inversions with low barriers interfragment coordinates in loose clusters etc should be kept to a minimum Similarly in transition states as few coordinates as possible should be combined to form the transition vector It is difficult to make more specific statements about good coordinate systems versus bad but often symmetry and the judicious choice of dummy atoms can be used to improve a coordinate system Section 5 4i As far as possible natural valence coordinates bond lengths and angles should be us
27. hoice of Internal Coordinates Although a good geometry optimization program should converge regardless of the coordinate system most practical optimization problerns converge much faster when the coordinate system is constructed with some care Strong coupling between coordinates invariably causes difficulties for optimizations The coupling between stiff coordinates stretch bend etc and loose coordinates e g internal rotations and inversions with low barriers can be especially troublesome Cyclic molecules have inherently strongly coupled internal coordinates and suffer from additional difficulties because of redundant coordinates Transition structures are sometimes quite flexible and coupling between these flexible modes and the transition vector may lead to additional problems Loosely bound clusters are a third category of potentially troublesome optimizations because large changes in geometry may occur with small changes in the energy Internal coordinates for the GAUSSIAN system of programs are defined using the Z matrix notation 4 other programs use similar systems to define internal coordinates see 29 and the appropriate user manual should be consulted for details To facilitate the discussion of internal coordinate systems an example of a Z matrix input for methanol is given below For the first atom C only the name of the atom is required for atom 2 the name of the atom O the number of the atom to which it is bonded 1
28. iRI C1RI1 C2R 1 108 O2R21A2 O 2 R2 1 A2 C3R2108 10 C3R22A310 C3R32A310 C4R3 108 20 C4R13A220 C4R43 A420 Cs Cy Cs Cy Cs Ny ea Ca C2 C4 C C Cc N 2 4 Ce c3 ac al oo Example 9 Less coupled coordinate systems for 5 membered rings a C5 D5h symmetry b C40 Coy symmetry c C3NO Cs symmetry Xx X X X 11 0 O1RI1 X 11 0 C1R290 C1R22 90 C1R12 90 C1R290 3 72 C 1 R2290 3 180 N 1 R1 2 90 3 180 C1R2 90 3 144 C3R31A3 20 C 3 R2 1 A2 2 180 C 1 R 290 3 216 C4R31A3 20 C 4 R3 1 A3 2 180 C 1 R290 3 288 O 1 R4 3 A4 2 180 X2 O2 O7 C7 Ce yY Ce Cs Ce Cs d X Cg C4 X C3 N4 X1 C3 C5 4 X2 Loose complexes can also cause problems for optimization programs in part because of the large changes in geometry that often occur during their optimization It is important that the 2 or more molecules in the complex have the necessary rotational and translational freedom relative to each other to the extent allowed by the symmetry of the complex as shown in Example 10 Furthermore the coordinates for the relative motion should not be coupled to any other coordinates Even with these precautions it may be necessary to freeze the internal degrees of freedom and optimize the interfragment modes first before optimizing all coordinates of the complex at the same time 43 Example 10 Symbolic Z matrix for the SIHF HCI cluster Si 1 Hs H 1 RH l F IRF 2AF i H CIIRCL
29. in recent years involve geometry optimization to some degree 2 3 In this Chapter some of the practical aspects of geometry optimization are considered The discussions are based on the GAUSSIAN series of programs 4 the optimization tools available in the GAUSSIAN system are listed in the Appendix but the concepts and suggestions can be applied directly to any semi empirical or ab initio MO program that uses internal coordinates for geometry optimization Most molecular mechanics or empirical force field programs carry out the optimization in Cartesian coordinates and hence are outside the scope of this Chapter Geometry optimization has become routine primarily because of the availability of efficient programs to calculate analytical energy gradients for recent reviews see 5 8 Analytical gradient based geometry optimizations are almost an order of magnitude faster than optimization methods that use only the energy Almost all user friendly molecular orbital programs have analytical gradients at the Hartree Fock HF level many have analytical gradients for the second order Mgller Plesset perturbation theory MP2 Some 33 J Bertr n and 1 G Csizmadia eds New Theoretical Concepts for Understanding Organic Reactions 33 53 1989 by Kluwer Academic Publishers 34 programs can also compute analytical gradients at the MP3 9 MP4 10 configuration interaction 11 coupled clusters 12 and MCSCF 13 levels Many semi e
30. l modes will retain the symmetry of the molecule displacement along any of the non totally symmetric modes will distort the molecule to a lower symmetry Thus the number of degrees of freedom that must be included in a full optimization of a molecule within a given symmetry is equal to the number of totally symmetric vibrational modes If character tables are not available an alternative scheme can be constructed to count the number of coordinates required in a full optiraization All the information needed is contained in the framework group notation for the molecule 33 Each symmetry equivalent set of atoms dummy atoms excluded contributes the smallest of the following to the number of degrees of freedom a 0 if an atom in the set is at the center of inversion the intersection of two axes of rotation or the intersection of a Gp plane and an axis b 1ifan atom in the set is on a proper or improper rotation axis c 2if an atom in the set is in a symmetry plane d 3if not in any symmetry element To account for translational and rotational invariance the smallest of the following is subtracted from the above sum to yield the number of internal coordinates that must be optimized in a full optimization a 0 if there is an intersection of two axes of rotation Dn Dah Dag and the cubic groups b 1 if there is a rotational axis and a symmetry plane Cnv Cnh c 2 if there is only a rotational axis Cn Sn d 3 if t
31. le has any symmetry it should be used to reduce the number of coordinates that must be optimized This can be beneficial even if the molecule has only Cs or C2 symmetry Taking advantage of symmetry may also fix some flexible coordinates such as internal rotations that would otherwise slow down the optimization In favorable circumstances symmetry can even be used to turn a transition structure optimization into a minimization see below Symmetry can also cause problems By symmetry the gradients must belong to the totally symmetric representation of the point group of the molecule This means that a gradient based optimization method will not change the symmetry of the molecule during the course of an optimization provided the internal coordinates properly reflect the symmetry of the molecule Hence a molecule will not distort to a lower symmetry during 35 an optimization even if the lower symmetry structure is lower in energy Therefore once the structure has been optimized it must be tested to ensure that displacements to lower symmetries do not lower the energy see below 12 Testing Stationary Points By contrast there are no restriction on optimizations going to higher symmetry however such optimizations may be slow because of strong coupling between symmetry equivalent coordinates Under such circumstances it may be unclear if the optimization will converge to a slightly distorted structure or if the optimization is actually
32. may be necessary in a few of the most difficult cases 12 Testing Stationary Points Any stationary point found in an optimization should be tested to be sure that it has the proper number of imaginary frequencies or negative eigenvalues of the Hessian i e 0 for a minimum and 1 for a transition state or first order saddle point This is done by computing the full Hessian either analytically or numerically and diagonalizing the matrix or computing the vibrational frequencies Note that the approximate Hessian obtained by an updating procedure in an optimization is not sufficiently accurate to test a stationary point Furthermore it does not contain any information about displacements to lower symmetry structures If the stationary point has the wrong number of negative eigenvalues 50 a lower energy stationary point with the right number of negative eigenvalues can be found by distorting the molecule along the offending eigenvector and re optimizing Often this leads to a structure with lower symmetry and requires the Z matrix to be rewritten in a lower symmetry form If the potential energy surface is not too flat it has been common practice to test a stationary point at one level of calculation and assume that the addition of more basis functions and or electron correlation does not change the nature of the stationary point Weakly bound complexes very flexible transition states and structures with strong configurational mixing can provide
33. mpirical programs also have analytical gradients In addition to first derivatives of the energy some calculations also require second derivatives e g vibrational frequencies Second derivatives can be calculated by numerical differentiation of analytical first derivatives or more efficiently by analytical second differentiation of the energy 14 A number of programs can calculate analytical second derivatives at the Hartree Fock level a few can calculate analytical second derivatives at the MP2 15 CI 16 and MCSCF 17 levels Third derivatives can also been calculated analytically at the Hartree Fock 18 and two configuration SCF levels 19 Additional details of the theory of analytical derivatives can be found in various articles and reviews 5 8 20 22 and references cited therein Details of the various algorithms used for geometry optimization and transition structure searching can be found in recent reviews 23 25 and will not be covered in the present Chapter From a numerical analysis point of view geometry optimization is just a problem in unconstrained minimization 26 28 The books by Scales 26 and Fletcher 27 are quite readable and discuss a wide variety of algorithms for unconstrained minimization with special emphasis on methods employing gradients Locating a transition structure is somewhat more difficult than finding a minimum A transition structure is a maximum in one and only one direction on the potential ene
34. nd some remedies are suggested however there are no guaranties that the remedies will work Forces too large While this is not normally a problem it may signal an error in the input a the starting geometry may be poor or b the coordinate system may be badly chosen PAE the E erea to De a poor geometry Either get a better starting guess for the geometry or reconstruct the Z matrix to avoid the strong coupling that optimization to take a bad step i kemp TRO Negative eigenvalues of the Hessian during a minimization This indicates that a the structure is not a minimum or b numerical problems occurred in updating the Hessian For a displace along the offending eigenvector to get to a lower energy structure and continue the optimization For b restart the optimization with a better estimate of the Hessian If the problem persists it may be necessary to freeze the coordinates that dominate the vector with the negative eigenvalue When the remaining coordinates have converged the frozen variables can be released so that all coordinates can be optimized simultaneously Too many negative eigenvalues of the Hessian during a transition structure optimization Either a the optimization is converging on a second order saddle point or b there are numerical problems with the Hessian In case a examine the eigenvectors with the negative eigenvalues and choose one that does not correspond to the transition vector for the desired te
35. ne for the identity SN2 reaction X CH3X XCH3 X The effect of c for these types of reactions is to constrain the transition state to be at the midpoint of a symmetric reaction path connecting reactants and products Since the reaction path is the direction along which a maximum must be found and the position of this maximum given by symmetry what remains to be done in the transition structure optimization is to minimize with respect to all of the remaining totally symmetric displacements in the transition state Thus for these special cases the transition structure optimization is reduced to a simple minimization The final transition structure should still be tested to ensure that it is a valid transition state i e with one and only one negative eigenvalue in the full Hessian Section 12 3 The Basic Optimization Step Most of the algorithms for geometry optimization using gradients rely on a quadratic expansion of the energy surface 26 28 In terms of the coordinates Xo the calculated energy Eo the calculated gradient go and an approximate second derivative matrix or Hessian H the energy and gradient are written as E Eo g X Xo 1 2 X Xo H X Xo 1 Bo H X Xo 2 At the optimized geometry the gradient is zero therefore the next estimate of the optimum geometry is found by solving eq 2 for g 0 i e a Newton Raphson step 36 X Xo H o 3 Almost all of the algorithms have th
36. nimizations 49 No empirical force field is sufficiently general and reliable for transition states c Semi empirical Hessians Hessians calculated by semi empirical MO methods are generally quite reasonable Usually the Hessian must be scaled if it is used for ab initio calculations since semi empirical methods over estimate some terms and underestimate others A bit of caution is necessary for transition structures because the geometry and hence also the Hessian for some transition structures optimized by semi empirical methods can be rather different from those computed by ab initio methods d Numerical calculation of key elements of the Hessian Gradient calculations at small displacements from the initial geometry can be used to calculate the more important rows and columns of the Hessian For transition structure optimizations it is essential that these include the coordinates that dominate the transition vector e Calculation of the full Hessian The full Hessian can be calculated analytically or by numerically differentiating the gradients The Hessian can be calculated at the same level as the optimization or with a smaller basis set or at the SCF level for an optimization with correlation Alternatively but less accurate the approximate Hessian from a lower level optimization on the same structure can be used f Recalculation of the full Hessian at each step in the optimization This is the most expensive option but
37. or b the molecule has Cs or C2 symmetry and the principle axes have changed order continue the optimization possibly with symmetry suppressed Conclusions This Chapter has attempted to discuss some of the practical problems of geometry optimization Though optimization algorithms vary from MO program to MO program they share a number of features and shortcomings The 3 areas of input that most affect the performance of a given geometry optimization algorithm are a the choice of internal coordinates redundancy and strong coupling must be avoided b the starting geometry and c the initial estimate of the Hessian Symmetry the use of dummy atoms coordinate redundancy strong coupling among coordinates conversion between coordinate systems and testing of stationary points have been considered in some detail Finally some of the conditions that cause optimizations to misbehave have been discussed and some remedies have been suggested Although much of the discussion draws on experience with the Sl GAUSSIAN series of programs the concepts and suggestions should be applicable to most geometry optimizations and transition structure searches based on internal coordinates Appendix Optimization Searching and Numerical Differentiation Links in GAUSSIAN L101 Symbolic Z matrix input L102 Fletcher Powell minimization method using only the energy 56 L103 fae optimization method for equilibrium geometries and transition struc
38. r molecules to avoid valence angles of 180 molecules undergoing inversion so that the entire path from reactants to products can be followed with the same Z matrix cyclic molecules to uncouple coordinates that are too strongly coupled and transition states to help separate the transition vector from the remaining coordinates To permit geometry optimization in GAUSSIAN the values for distances and angles to be optimized are replaced by variable names Initial values are supplied for these parameters and the optimization code varies these parameters in the search for the stationary point There are numerous other features in the GAUSSIAN Z matrix input 4 but the limited set illustrated here is sufficient for the examples discussed below Example 2 Symbolic Z matrix for methanol HF 3 21G optimized values c H4 O i RCO H 2 ROH 1 ACOH H 1 RCHI 2 AOCHI 3 180 Ci O H 1 RCH2 2 AOCH2 4 DHOCH2 tig H 1 RCH2 2 AOCH2 4 DHOCH2 i a 5 RCO 1 440 ROH 0 966 RCH1 1 079 RCH2 1 085 ACOH 110 4 AOCH1 106 3 AOCH2 1 12 3 DHOCH2 118 5 Most programs impose some restriction on the value of distances and angles used to construct the geometry Distances are usually required to be positive valence angle must be greater than 0 but less than 180 and dihedral angles typically have a range of 180 to 180 or 360 to 360 Other restrictions may be imposed by the symmetry of the molecule e g if an angle is tetrahedral by symmet
39. rgy and is a minimum in all other directions i e a first order saddle point or col During the course of a transition structure search the algorithm must choose the best direction along which the energy is maximized as well as carrying out the maximization in that direction and the minimization in all other directions Compared to unconstrained minimizations the numerical analysis literature on saddle point optimizations is much less extensive 26 28 and references cited From a chemical point of view a number of methods for finding transition states have been proposed details of some of these algorithms can be found in recent reviews 23 24 and references therein Most programs provide a brief write up of how to use their particular optimization code A useful outline for geometry optimization can also be found in A Handbook of Computational Chemistry 29 However there does not seem to be a practical guide on the details of geometry optimizations and transitions structure searches The purpose of this Chapter is to provide a few hints to help set up optimizations so that they will work better Sections 4 12 and to provide some suggestions of what to try when optimizations go wrong Section 13 2 Symmetry and Stationary Points Symmetry can be quite helpful in speeding up optimizations of minima and saddle points but can also create difficulties by constraining the search to a subsection of the energy surface If a molecu
40. ry 109 4712 should be used and not 109 5 Some care must be taken that the limits on the bond lengths and angles are not exceeded during the course of an optimization For example a molecule with 3 atoms nearly linear such as HOCN or any molecule with a fragment that can approach linearity during an optimization can be specified with a dummy atom and thus avoiding the restrictions on the valence angle 38 Example 3 Symbolic Z matrix for cyanic acid HF 3 21G c X N 1 RCN X 110 2 90 O 1 RCO 3 AXCO 2 180 hate H 4 ROH 1 ACOH 3 180 RCN 1 140 RCO 1 308 ROH 0 970 Hs AXCO 88 7 ACOH 114 2 5 Internal Coordinates Symmetry and Dummy Atoms Some useful guidelines for constructing internal coordinate systems can be developed for molecules with some degree of syinmetry 32a Specifically symmetry can be quite helpful in setting up dummy atoms and in defining the connectivity for the internal coordinates For example cyclic molecules can lead to very strongly coupled coordinate systems but if there is sufficient symmetry a well behaved relatively uncoupled coordinate system can be constructed Even if a molecule possesses very little symmetry a suitable set of internal coordinates may be obtained by taking a higher symmetry analogue of the molecule and systematically reducing the symmetry For molecules with more than 3 or 4 atoms and with some symmetry it may be best to define the atoms on the highest symmetry elem
41. tures L105 Murtagh Sargent minimization method using gradients 57 L106 Calculation of the Hessian by numerical differentiation of the gradients L107 Linear synchronous transit requires only the energy 39 L108 Potential surface scan L109 Fixed metric optimization for equilibrium geometries and transition structures using gradients obtained by numerical differentiation of the energy L110 Calculation of the Hessian by numerical second derivatives of the energy L111 Calculation of hyperpolarizabilities by numerical second derivatives of the dipole moment L113 Eigenvector following algorithm using numerical gradients 42 L114 Eigenvector following algorithm using analytical gradients 42 L115 Reaction path following using gradients 55 References 1 Hehre W J Radom L Schleyer P vR and Pople J A Ab Initio Molecular Orbital Theory Wiley Interscience New York 1986 Qs Ohno K and Morokuma K Quantum Chemistry Literature Data Base Elsevier Amsterdam 1982 yearly supplements published in special issues of the journal J Mol Struct Theochem on line version available through Japan Assoc for International Chemical Information 3 Whiteside R A Frisch M J and Pople J A The Carnegie Mellon Quantum Chemistry Archive 3rd Ed Carnegie Mellon University Pittsburgh 1983 l current version available on line from Gaussian Inc 4 Frisch M J Binkley J S DeFrees D J Raghavach
42. ummy atom is placed on the C3 axis then the variable BXCH can define the entire inversion process smoothly Example 7 Symbolic Z matrix for out of plane bending in CH3 a potential problems b better coordinates X2 C Cc H H 1 RCH X110 53 H 1 RCH 2 AHCH H 1 RCH 2 BXCH C1 H3 H 1 RCH 2 AHCH 3 AHCH1 H 1 RCH 2 BXCH 3 120 H 1 RCH 2 BXCH 3 240 H4 Cyclic molecules are considerably more difficult to specify than acyclic molecules primarily because the natural valence coordinates have unavoidable redundancies For example a planar 5 membered ring has 5 bonds and 5 valence angles for a total of 10 in plane coordinates however there are only 7 degrees of freedom to optimize Cs symmetry Thus each valence coordinate is necessarily strongly coupled to the others Examples 8 and 9 illustrate 2 coordinate systems for each of three model planar rings C5 Dsn e g the heavy atom skeleton for C5Hs57 C40 Cav the skeleton for furan and C3NO Cz the skeleton for isoxazole If a ring is specified as a long chain of atoms then a change in any one of the bonds or angles will change the length of the ring closure bond thus resulting in a very strongly coupled coordinate system The second coordinate system for each case is less strongly coupled or for Cs makes better use of symmetry 42 Example 8 Strongly coupled coordinate systems for 5 membered rings a C5 D5h symmetry b C40 C2y symmetry c C3NO Cs symmetry C C N CIR C
43. unfreeze the coordinates and optimize everything d there is strong coupling in the coordinate system restructure the Z matrix and restart the optimization or e the initial estimate of the Hessian was quite poor recalculate a few of the elements of the Hessian and continue the optimization Maximum step size exceeded If the step size is larger than the trust radius most optimizers scale the step to the appropriate length If this occurs frequently it may be indicative of a small eigenvalue in the Hessian the causes and remedies for this have been discussed above Step size too small optimization goes nowhere despite sizeable gradients This could be due to a too small a trust radius b tightly coupled coordinates and or a very non quadratic energy surface or c a Hessian with some spuriously large matrix elements Case a is probably caused by inappropriate updating of the trust radius hence this feature should be temporarily disabled and the optimization continued The remedy for b is to reconstruct the Z matrix to avoid the strong coupling and to simplify the behavior of the energy surface Case c can be overcome by continuing the optimization with an improved estimate of the Hessian Change in point group detected during an optimization Either a the Z matrix does not reflect the full symmetry of the molecule and the optimization has inadvertently distorted the molecule fix the Z matrix and restart the optimization
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