Volume 182, number 5 CHEMICAL PHYSICS LETTERS 9August 1991
Transition-state optimizations by trust-region image minimization
Trygve Helgaker
Departmenl ojChemisvy, Unrvers~fy of OS/o, P.O. Box 1033 Blindem, MI315 Oslo 3. Norwyv Received 3 May 199 I ; in final form 20 May 199 1
A new method for optimizing transition states is presented. The method combines Smith’s image function with trust-region minimization. Calculations on HCN and C2H, illustrate the usefulness of the method for ab initlo potential energy surfaces. It is found that second-order image optimizations of transition states are as fast as conventional minimizations.
1. Introduction
It was recently suggested by Smith that in order to optimize a transition state, we should minimize the image of the potential energy surface [ 11. The image of a surface is a function whose gradient and Hessian at each point are identical to those of the original surface except for opposite signs in the lowest eigen- mode. The minima of the image, therefore, coincide with the transition states of the energy surface and we may determine transition states by minimizing the image.
Smith proposed a first-order method for image minimizations of transition states [ I 1. However, there is no reason we should restrict ourselves to first- order methods since the Hessian is needed anyway in order to identify the mode to be reflected (the im- age mode). In this paper, the image function is com- bined with second-order trust-region minimization
[ 21. The resulting trust-region image minimization (TRIM) method is applied to HCN and CzH6. These calculations suggest that transition-state optimiza- tions using TRIM are as fast as conventional minimizations.
2. Trust-region image minimization
LetS(x) be the function to be optimized. The gra- dient and Hessian at x are given by
g(x) =Vf(x) , (1)
G(x)=Vy”(x).
(2)
Diagonalizing the Hessian, we obtain eigenvectors V~
and eigenvalues i k, GUN =lkvfi,
and lhe gradient may be written as
(3)
g= c @Auk. (4)
1.
In the diagonal representation, the image function s(x) has the same gradient and Hessian asl(x) ex-
cept for opposite signs in the lowest mode:
image or reaction mode,
&=,&, &k=&,
(5)
transverse modes (k# 1) . (6)
Therefore, the minima off(x) coincide with the first- order saddle points of f(x) and vice versa. This means that we may determine the saddle points of S(x) by minimizing f(x).
Although we only consider image functions ob- tained by reflecting the lowest mode, we may in prin- ciple reflect any mode. At each step, we must then identify the image mode by comparing old and new eigenvectors. We avoid this by always selecting the lowest mode.
In the trust-region method, we minimize the sec- ond-order Taylor expansion off(x) around the cur-
0009-2614/91/$ 03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland) 503
Volume 182, number 5 CHEMICAL PHYSICS LETTERS 9 August 199 1
rent estimate of the minimizer xc within the trust region
minM(x)=f,+&~t~~ G,Ax subject tozb<h2,
where
(7)
Ax=x-x, . (8)
Using Lagrange’s method for unconstrained min- imization, we obtain
A0)=-(G,-n)-‘g,> (9)
where )I is an undetermined multiplier. The Newton step (p= 0) is taken if x( 0) is smaller than the trust radius h. Otherwise, we adjust p< 0 such that the step is to the boundary,
I Wru) I =h. (10)
In the diagonal representation, the step may be writ- ten as
and the diagonal components are obtained from the equations,
&=- -!!5_
&i-P”’ (12)
The trust radius h is initially given some arbitrary but reasonable value. During the optimization, h is updated based on comparisons made between the actual and predicted changes in the function.
This algorithm requires only minor changes to carry out minimizations of the image function. Let us write the second-order expansion of the function in eq. (7) in the diagonal representation,
M(x) =f, + 5 m/J&) 3 (13)
where
mk(ik)=Q&+f&iZ. (14)
The corresponding expansion of the image function is
M(x) =L + ; %(ik) 3 where
504
(15)
Mik,=&:k+fJkG
3(16)
and L is unknown. From eqs. (5) and (6), we see that
fil(~,)=-m,(6,), imagemode,
&(ik)=mk(ik),
transverse modes (k# 1) , (17)
and the second-order expansion of the image func- tion may be written as
M(x)=L-ml(&)+ Jl mk(ik). (18)
Therefore, the step bx on the image function,
AX= 1 ikvk > (19)
k
may be calculated as
[,+$=-~;
image mode,transverse modes (k# 1) , (20) where p is adjusted such that Ax IS to the boundary of the trust region. The only difference between the image step and the transverse steps is the sign of the level shift. The level shift is added to the image ei- genvalue and subtracted from the transverse eigenvalues.
Notice that minimization of the image function corresponds to a simultaneous minimization of transverse modes and maximization of the image mode. This is achieved by changing the sign of the second-order contribution to the image mode, eq.
( 18). Increases in the image mode and reductions in the transverse modes are weighted equally.
3. Sample calculations
To test the TRIM method, we have carried out minimizations and transition-state optimizations on HCN and C2H6 at the ab initio Hartree-Fock level.
All calculations were carried out using the SIRIUS/
ABACUS program system [ 3,4].
Volume 182, number 5 CHEMICAL PHYSICS LETTERS 9 August I99 1
3.1. Calculations on
HCN
In the calculations on HCN, we used the MINI-3 basis set of Tatewaki and Huzinaga [ 5 1. This mol- ecule has two linear minima (HCN and HNC) and
a transition state connecting these minima, see table 1. We carried out three sets of calculations starting at different geometries in the global region, see tables 2-4. The geometries were chosen fairly randomly with the bonding angle HNC in the range 14-154”.
Table I
HCN stationary points
HCN (Cm, 1 HNC (Cm,) TS (‘2
energy (au) Rh.c (A) RCH (A)
&I (A) L CNH
dipole moment (D) o+/a’ (cm-l) x/a’ (cm-‘)
-92.675839 1.1391 1.0522 linear
3.13 3664 2364 898
- 92.662634 1.1597 0.9824 linear
2.92 4020 2244 666
-92.574521 1.1821 1.2186 1.4036 55.44”
1.52 2447 2132 1163i
Table 2
(A) Optimization of HCN minimum a). (B) Optimization of HCN saddle point b’. Initial geometry: R,c=0.7938 A, R,c=0.3742 A, LNCH= 165.96”. In au
Iteration E ” Gradient d, Index Cl &in r’ RTS g’ Size ” Shift ‘) Ratio”
A 1 2.8330 9.5544
2 1.2409 4.4064
3 0.5069 1.8560
4 0.2169 0.7234
5 0.1079 0.2669
6 0.059 I 0.1142
7 0.0247 0.0755
8 9 IO 11
0.0061 0.0005 0.0000 0.0000
0.0550 0.0380 0.0013 0.0000
B 1
2 3 4 5 6 I 8 9 ID
2.1317 9.5544 0.9128 3.7701 0.2099 1.3825 0.0038 0.4749 -0.0217 0.1762 - 0.0063 0.0943
0.0019 0.0000 0.0000 0.0000
0.0750 0.0090 0.0001 0.0000
I
1.55 1.30
1.59 1.01
1.73 0.78
1.53 0.59
1.31 0.60
1.02 0.85
0.65 1.23
0.31 1.58
0.04 1.83
0.00 1.83
0.00 1.84
1.55 1.30
1.26 1.14
I .04 1.03
0.99 0.93
1.20 0.69
1.51 0.36
1.89 0.08
I.83 0.01
1 .a4 0.00
1.84 0.00
0.31 20.9 0.82
0.31 5.49 0.97
0.32 1.32 1.07
0.32 0.67 1.01
0.34 0.42 0.92
0.40 0.24 0.90
0.35 0.12 0.94
0.29 Newton 1.01
0.04 Newton 0.99
0.00 Newton I .oo
0.36 0.64 1.25
0.35 Newton 1.25
0.27 Newton 1.21
0.27 0.22 1.10
0.33 0.21 0.89
0.40 0.03 0.83
0.08 Newton 0.95
0.01 Newton 1.01
0.00 Newton 1 .oo
a) Total length of walk: 2.69. Detour ratio: 1.73 (see footnote I ). b, Total length of optimization: 2.07. Detour ratio: 1.59.
‘) Energy relative to minimum. d, Norm of gradient in atomic units. ‘) Number of negative Hessian eigenvalues.
r) Distance to minimizer. I1 Distance to minimizer of image surface. h’ Size of step.
‘) Negative level shift parameter. j) Ratio between calculated and predicted change in energy for step.
Volume 182. number 5 Table 3
CHEMICAL PHYSICS LETTERS 9 August I99 I
(A) Optimization of HNC minimum ‘). (B) Optimization of HNC saddle point bl. In au ‘)
Iteration E Gradient Index R ,,,>n RTS Size Shift Ratio
A I
4
8
10 11 12 13
B 1
2 3 4
8
10 I1
5.0799 22.633
1.8320 8.8338
0.6454 3.2817
0.2207 1.1252
0.0935 0.3136
0.0564 0.0752
0.0337 0.0657
0.0149 0.0574
0.0045 0.0379
0.0006 0.0289
0.0000 O.OOll
0.0000 0.0001
0.0000 0.0000
4.99 18 22.633
1.4820 7.8092
0.3885 2.7387
0.0516 0.8775
-0.0109 0.2190
-0.0139 0.0646
-0.0086 0.0410
-0.0029 0.0329
0.0001 0.0182
0.0000 0.0003
0.0000 0.0000
I 1.74
1 1.88
1 2.10
1 1.90
I 1.68
I 1.40
0 1.06
0 0.68
0 0.36
0 0.08
0 0.02
0 0.00
0 0.00
1 1.74
1 1.58
1 1.50
1 1.52
1 1.65
1 1.75
I I .88
I 2.03
1 2.20
1 2.22
I 2.22
1.15 0.31 26.5 0.94
0.88 0.31 6.03 I .04
0.65 0.31 1.70 I .06
0.57 0.32 0.3 I I.1 I
0.63 0.35 0.24 0.97
0.87 0.39 0.18 0.90
1.22 0.40 0.10 0.97
1.61 0.34 0.05 I .03
1.92 0.29 Newton 1.07
2.18 0.06 Newton 1.04
2.20 0.02 Newton I .oo
2.22 0.00 Newton I .oo
2.22
1.15 0.32 Newton 1.24
0.94 0.29 Newton 1.25
0.79 0.22 Newton 1.22
0.71 0.19 Newton I.15
0.58 0.11 0.42 ,J’ 1.06
0.50 0.13 0.29 I .O?
0.38 0.16 0.20 0.98
0.23 0.19 0.06 0.86
0.04 0.00 a.00
0.04 Newton 1.05
0.00 Newton 1.00
a) Total length ofoptimization: 3.09. Detour ratio: 1.78. b’ Total length of optimization: 1.66. Detour ratio: 1.44
‘) For column headings and abbreviations, see table 2. d, First step from this geometry was rejected.
The energies in the first iteration were from 1 to 5 au above the converged energies, with gradients in the range 4-23 and one negative eigenvalue.
From each geometry, we carried out one minim- ization and one transition-state optimization, The optimizations were converged to a gradient norm smaller than 1 O-4. About ten steps are needed in each case, three of which are Newton steps in the local re- gion. The method performs equally well for mini- mum and transition-state optimizations.
The first set of calculations is listed in table 2. The minimization converges in ten steps with a detour ratio ” of 1.73, and the saddle point is found in nine
*’ The detour ratio is the ratio between the total length of the walk and the distance from the initial to the final point. A de- tour ratio of one means that the optimization goes in a straight line whereas higher ratios indicate that the optimization is indirect.
506
steps with a detour ratio of 1.59. In each case, four steps are in the local region.
Table 2 also gives the distances from the current geometry to the minimum (R,,,) and to the tran- sition state ( RTS). Table 2 (A) shows that in the first two iterations of the minimization, the distance to the minimizer increases, which explains the rather high detour ratio. It appears that the minimization is being distracted by the saddle point since, in it- eration, 4RTs is much smaller than R,,,. In spite of this, the gradient decreases in each iteration and the algorithm has no difficulties in locating the mini- mum.
The transition-state walk is slightly more direct but note that in iteration 4, the geometry is halfway be- tween the minimum and the transition state. In it- erations 2 and 3, the Newton step is taken although we are still in the global region. This does not happen
Volume 182, number 5 CHEMICAL PHYSICS LETTERS 9 August I99 I Table 4
(A) Optimization of HNC minimum a). (B) Optimization of HNC saddle point b). Initial geometry: R,c=0.7938 A, R,,=0.5916 A, LHNC=153.44”.Inauc’
Iteration E Gradient Index R m,n RTS Size Shift
A 1 1.1489 3.9115 1 1.09 1.58 0.35 5.13
2 0.402 I 1.7387 1 1.05 1.33 0.32 1.51
3 0.1212 0.6097 1 1.17 1.09 0.35 0.46
4 0.0370 0.1800 1 0.90 1.33 0.38 0.19
5 0.0103 0.0393 0 0.56 1.69 0.35 0.06
6 0.0021 0.0343 0 0.23 2.02 0.18 Newton
I 0.0001 0.0113 0 0.05 2.16 0.05 Newton
8 0.0000 0.0008 0 0.01 2.19 0.0 1 Newton
9 0.0000 0.0000 0 0.00 2.20
B 1 1.0608 3.9175 I 1.09 1.58 0.35 0.97
2 0.2434 1.4696 1 0.78 1.61 0.33 Newton
3 -0.0285 0.4274 1 0.55 1.68 0.22 Newton
4 -0.0752 0.0840 0 0.56 1.67 0.32 0.11
5 -0.0647 0.0600 0 0.81 1.40 0.32 0.21
6 -0.0480 0.0713 0 1.17 1.09 0.38 0.18
7 -0.0262 0.0825 1 1.52 0.74 0.39 0.14
8 -0.0073 0.0728 1 1.88 0.41 0.42 0.09
9 0.0057 0.0915 1 2.26 0.16 0.13 Newton
10 0.0000 0.0181 1 2.16 0.07 0.07 Newton
11 0.0000 0.0012 1 2.20 0.00 0.00 Newton
12 0.0000 0.0000 1 2.20 0.00
a1 Total length of optimization: 1.98. Detour ratio: 1.82. b1 Total length of optimization: 2.93. Detour ratio: I .85.
‘) For column headings and abbreviations, see table 2.
Ratio 0.94
I .oo 0.97 0.85 0.86 1.14 I .06 1.01
1.19 1.21 1.12 1.32 0.96 0.94 0.94 0.85 1.13 1.32 1.01
Table 5
Staggered and eclipsed conformations of ethane
energy (au) Rcc (A) RCH (A) LHCC (deg) LHCH (deg) as/a; (cm-‘)
e,/e’ (cm-‘)
a,,/a; (cm-‘) a2Ja; (cm-‘) e,/e” (cm-‘)
Staggered ( Djd) Eclipsed (D3h)
-79.115933 -79.111519
1.5288 1.5411
1.0834 1.0825
111.2 111.6
107.7 107.3
3185 3199
1584 1607
1053 1046
3230 3239
1665 1664
1360 1322
314 305i
3179 3185
1587 1585
3258 3264
I669 1677
924 1001
in the early stages of the minimization since the Hes- sian is indefinite. The relatively poor ratio between the actual and predicted changes in the energy in the first iterations indicates that this region is highly an- harmonic. In iterations 5 and 6, the energy is slightly lower than at the transition state. Still the gradient decreases steadily during the optimization.
The second set of calculations (table 3 ) shows the same behavior. The initial geometry is closer to the transition state than to the minimum (HNC) which perhaps explains the higher detour ratio of the min- imization. Again, the minimization first moves away from the minimum and towards the saddle point whereas the image optimization starts out with four consecutive Newton steps, which slows the optimi- zation down. However, in both cases the gradient de- creases steadily throughout the optimization% re- flecting the stability of the method.
The third set of calculations (table 4) starts closer
Volume 182, number 5 CHEMICAL PHYSICS LETTERS 9 August 199 1 Table 6
Optimization of ethane. (A) Staggered conformation ai. (B) Eclipsed conformation ‘). Initial geometry: carbons at (0, 0,O) and (3,0, O),hydrogensat (-1,0.8,0.2), (-I, -0.5, -0,4),(-I, -0.4,0.5), (4,0.5, -0.4), (4, -0.7,0.2), (4.0.5,0.4).lnau”
Iteration E Gradient Index R mln R TS Size Shift Ratio
A 1 3.221 I
2 I .6038
3 0.7941
4 0.3876
5 0.1752
6 0.0627
1 0.0125
8 0.0003
9 0.0000
10 0.0000
B I 3.2167
2 1.5594
3 0.7505
4 0.3590
5 0.1568
6 0.0505
7 0.0057
8 -0.0034
9 -0.0016
IO - 0.0002
II 0.0002
12 0.0000
13 0.0000
14 0.0000
5.9416 2.5916 1.0697 0.5117 0.2831 0.1456 0.05 13 0.0088 0.002 1 0.0000 5.9416 2.5455 1.0396 0.4939 0.2700 0.1345 0.0450 0.0077 0.0158 0.0150 0.0145 0.0016 0.0007 0.0000
7 3.2 I
7 2.94
7 2.55
4 2.07
3 1.59
1 1.15
1 0.74
0 0.24
0 0.02
0 0.00
7 3.21
7 2.93
7 2.50
3 2.01
3 1.54
1 I.10
1 0.72
0 0.74
0 1.34
1 I .93
I 2.52
1 2.30
1 2.43
1 2.42
3.40 0.48 3.658 1.05
3.15 0.57 1.251 1.06
2.83 0.59 0.639 1.07
2.49 0.57 0.465 1.02
2.21 0.54 0.268 1.00
2.02 0.54 0.098 0.99
1.92 0.55 0.012 0.96
2.19 0.22 Newton 1.01
2.40 0.02 Newton 1.01
2.42
3.40 0.50 3.164 1.07
3.09 0.58 1.132 1.07
2.78 0.59 0.625 1.06
2.46 0.57 0.45 I 1.01
2.20 0.55 0.248 I .oo
2.01 0.53 0.087 0.99
1.89 0.49 0.004 0.97
1.70 0.60 0.007 1.00
1.11 0.61 0.007 0.17
0.51 0.61 0.003 0.56
0.10 0.22 0.001 d’ 1.28
0.12 0.01 0.00
0.13 Newton 0.9 1
0.01 Newton l.D3
ai Total length of optimization: 4.08. Detour ratio: 1.27. b, Total length ofoptimization: 5.98. Detour ratio: 1.76.
c, For column headings and abbreviations, see table 2. d, First step from this geometry (the Newton step) was rejected
to the HNC minimum than to the transition state.
The minimization is, therefore, faster. Both optimi- zations proceed in a roundabout manner with large detour ratios 1.82 and 1.85. For example, the image minimization spends three iterations passing through the positive definite region of the surface. It is low in energy in iteration 4 but then moves up again un- til finally, three Newton steps are taken to the saddle point.
molecule at the Hartree-Fock 4-31G level [ 61 with no symmetry restrictions. Table 5 lists the converged staggered (minimum) and eclipsed (transition-state) conformations of ethane.
In conclusion, TRIM performs well on this sur- face. Indeed, there is little difference between the minimum and saddle-point optimizations.
3.2. Calculations on C2H6
The first set of calculations (table 6) starts far away in the global region. The energy is about 3 au above the optimized energies, the gradient is 5.9 and the Hessian index 7. Nevertheless, the minimization converges in nine steps (seven global and two local) with a detour ratio of only 1.27. The TRIM walk is more indirect (detour ratio 1.76 1. It first follows the minimization and goes through the positive definite region of the staggered conformation in iterations 8 and 9. It then moves up towards the saddle point and converges in two steps upon entering the local region.
To test the TRIM method on a higher-dimensional In the final calculations, we optimized the stag- surface, we carried out calculations on the ethane gered conformations starting from the eclipsed and 508
Volume 182, number 5 CHEMICAL PHYSICS LETTERS 9 August 199 1 Table I
Optimization ofethane. (A) Staggered conformation. (B) Eclipsed conformation b). Initial geometry: see table 6. In au ‘)
Iteration E Gradient Index R m,n RTS Size Shift
A 1 0.0044 0.0000 1 2.42 0.00 0.50 0.000
2 0.0041 0.0102 1 1.94 0.50 0.50 0.004
3 0.0030 0.0103 0 1.45 1 .oo 0.60 0.004
4 0.0015 0.0146 0 0.86 1.59 0.60 Newton
5 0.0004 0.0144 0 0.27 2.17 0.16 Newton
6 0.0000 0.0008 0 0.1 I 2.31 0.10 Newton
7 0.0000 0.0005 0 0.00 2.41 0.00 Newton
8 0.0000 0.0000 0 0.00 2.42
B I - 0.0044 0.0000 0 0.00 2.42 0.50 0.000
2 -0.0038 0.01 IO 0 0.50 1.94 0.50 0.008
3 - 0.0027 0.0112 0 I .oo 1.45 0.60 0.008
4 - 0.0009 0.0153 0 1.59 0.86 0.61 0.005
5 0.000 1 0.0148 1 2.18 0.26 0.15 0.005 d,
6 0.0000 0.0008 1 2.32 0.11 0.11 Newton
7 0.0000 0.0005 1 2.42 0.01 0.01 Newton
8 0.0000 0.0000 1 2.42 0.00
a1 Total length of optimization: 2.46. Detour ratio: 1.02. b1 Total length of optimization: 2.48. Detour ratio: 1.02.
c) For column headings and abbreviations, see table 2. d1 The first two steps calculated at this geometry were rejected.
Ratio 0.70 1.05 0.95
1.10 1.09 1.02 1.00
1.24 0.85 0.87 0.74 1.19 0.96 1.01
vice versa (table 7). These optimizations are quite similar. Both take eight iterations and are very direct (detour ratio 1.02). They nicely illustrate the simi- larities between trust-region minimizations and TRIM optimizations of saddle points.
4. Discussion
We have demonstrated that it is possible to opti- mize transition states efficiently by combining Smith’s image function with second-order trust-re- gion minimization. The resulting TRIM algorithm appears to be robust and converges equally fast to minima and saddle points. The algorithm is simple and introduces only one heuristic parameter (the size of the trust region).
The TRIM approach is similar to other methods for a priori optimization of transition states [7-lo].
These methods all work by the same principle: in- crease the energy in one direction (the reaction mode or transition mode) and decrease it in all other di- rections (the transverse modes). They require at each step the gradient as well as the Hessian. The Hessian
is used both to identify the reaction mode and to generate the step. However, TRIM is in some ways more satisfactory. It generates the step by optimizing a single function. In contrast, the rational function [ 81 and gradient extremal [ 93 methods generate the steps along the reaction mode and the transverse modes independently. Therefore, there is no natural relationship between the step lengths in the reaction mode and the transverse modes. Unlike the gradient extremal method [ 91 and the method described by Nichols et al. [ lo], TRIM works from any point on the surface. It does not require a preliminary min- imization in order to carry out the transition-state optimization. And unlike the rational function ap- proach, the TRIM method has a natural step-length control since it is based on the concept of a trust region.
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