^{1,a)}

### Abstract

The reaction path is an important concept of theoretical chemistry. We use a projection operator for the following of the Newton trajectory (NT) along the reaction valley of the potential energy surface. We describe the numerical scheme for the string method, adapting the proposal of a growing string (GS) by [Peters *et al.*,J. Chem. Phys.120, 7877 (2004)]. The combination of the Newton projector and the growing string idea is an improvement of both methods, and a great saving of the number of iterations needed to find the pathway over the saddle point. This combination GS-NT is at the best of our knowledge new. We employ two different corrector methods: first, the use of projected gradient steps, and second a conjugated gradient method, the CG+ method of Liu, Nocedal, and Waltz, generalized by projectors. The executed examples are Lennard-Jones clusters, and , and an N-methyl-alanyl-acetamide (alanine dipeptide) rearrangement between the minima and C5. For the latter, the growing stŕing calculation is interfaced with the GASSIAN03 quantum chemical software package.

The work was made possible through the financial support of the Deutsche Forschungsgemeinschaft, and through the computer time of the University of Leipzig. The author especially thanks Professor Dr. H.-J. Hofmann from the Institute of Biochemistry, University Leipzig, for giving him an introductory lesson into the model world of alanine “dipeptide.”^{36}

I. INTRODUCTION

II. NEWTON TRAJECTORIES

III. GROWING STRING METHOD

IV. IMPROVEMENT OF THE PROJECTED GRADIENT SEARCH BY THE CG+ METHOD

A. Lagrangian condition

B. Zero eigenvalues

V. EXAMPLES

A. Müller–Brown PES

B. Four-well potential

C. Lennard-Jones cluster

D. Lennard-Jones cluster and the CG+ method

E. Alanine dipeptide

VI. IMPLEMENTATION OF CG+ AND GAUSSIAN03 FOR THE GS METHOD

VII. DISCUSSION

### Key Topics

- Peptides
- 20.0
- Ground states
- 7.0
- Quantum optics
- 5.0
- Eigenvalues
- 4.0
- Interpolation
- 4.0

## Figures

GS method on Müller–Brown potential from minimum to minimum . The predictor step at is shown, and the corrector back to the searched RP (arrows). The desired number of nodes is . The Newton trajectory to the initial direction is included for comparison. The growing string follows the NT very well.

GS method on Müller–Brown potential from minimum to minimum . The predictor step at is shown, and the corrector back to the searched RP (arrows). The desired number of nodes is . The Newton trajectory to the initial direction is included for comparison. The growing string follows the NT very well.

Reaction path on the Müller–Brown potential from minimum to minimum . An 11-nodechain is obtained using the Newton projector. The convergence of the growing string method needs the calculation of 19 gradients. The search direction for the GS method is readjusted on the course.

Reaction path on the Müller–Brown potential from minimum to minimum . An 11-nodechain is obtained using the Newton projector. The convergence of the growing string method needs the calculation of 19 gradients. The search direction for the GS method is readjusted on the course.

Effort of the growing ends string method for the MB potential.

Effort of the growing ends string method for the MB potential.

Newton trajectory (dashes) and the growing ends string method (connected bullets) for a four-well potential including an initial chain, for comparison. (a) The fixed search direction between both minima is not well chosen; for the second part of the RP, see text. (b) The search direction for the GS method is readjusted on the course.

Newton trajectory (dashes) and the growing ends string method (connected bullets) for a four-well potential including an initial chain, for comparison. (a) The fixed search direction between both minima is not well chosen; for the second part of the RP, see text. (b) The search direction for the GS method is readjusted on the course.

Growing ends string method with 23 nodes for the potential (assuming argon). Shown is the energy profile. The left arrows indicate the two outmost right atoms which are mainly involved in the rearrangement. From top to bottom, the resulting curves for the corrector thresholds , 0.5, and 0.1 are shown. The SP at node 11 is well approximated.

Growing ends string method with 23 nodes for the potential (assuming argon). Shown is the energy profile. The left arrows indicate the two outmost right atoms which are mainly involved in the rearrangement. From top to bottom, the resulting curves for the corrector thresholds , 0.5, and 0.1 are shown. The SP at node 11 is well approximated.

Two Newton trajectories for a four-well potential with minima C5 and . Thin long dashes: NT to search direction between both minima. (It is not well chosen.) Short bold dashes: The search direction for the NT is well adapted to connect the C5 and minima.

Two Newton trajectories for a four-well potential with minima C5 and . Thin long dashes: NT to search direction between both minima. (It is not well chosen.) Short bold dashes: The search direction for the NT is well adapted to connect the C5 and minima.

Approximation of Newton trajectories for alanine dipeptide between the C5 and minima, see text. Method: projected and inverse gradient [Eq. (9)] dampened by . The search direction for the NT is the direction from start to finish, and after three steps the corresponding direction from former chain points to finish. Shown is the energy profile in a.u. (a) , , maximal 55 steps per node. (b) and is so small that 151 steps per node are used. (c) NT between the two SPs, , and for the upper, as well as for the lower curve, maximal 151 steps per node.

Approximation of Newton trajectories for alanine dipeptide between the C5 and minima, see text. Method: projected and inverse gradient [Eq. (9)] dampened by . The search direction for the NT is the direction from start to finish, and after three steps the corresponding direction from former chain points to finish. Shown is the energy profile in a.u. (a) , , maximal 55 steps per node. (b) and is so small that 151 steps per node are used. (c) NT between the two SPs, , and for the upper, as well as for the lower curve, maximal 151 steps per node.

Convergence history for curve (b) of Fig. 7 for the norm of the projected gradient. In the inlay, curves of nodes 2 to 23 count from top to bottom.

Convergence history for curve (b) of Fig. 7 for the norm of the projected gradient. In the inlay, curves of nodes 2 to 23 count from top to bottom.

Approximation of Newton trajectories for alanine dipeptide between the minima and C5, see text. Method: modified CG+ optimization. The search direction for the NT is the direction from start to finish, and after three steps the direction from former chain points to finish. The energy profile is shown in a.u. The lower curve is for (max 134 steps per node: which were used throughout) and the upper one is for . The energies of the SPs and of the intermediate minimum are included.

Approximation of Newton trajectories for alanine dipeptide between the minima and C5, see text. Method: modified CG+ optimization. The search direction for the NT is the direction from start to finish, and after three steps the direction from former chain points to finish. The energy profile is shown in a.u. The lower curve is for (max 134 steps per node: which were used throughout) and the upper one is for . The energies of the SPs and of the intermediate minimum are included.

Approximation of Newton trajectories for alanine dipeptide between the minima and C5, see text. Methods: projected and inverse gradient (thin points) starting at C5, and modified CG+ optimization (thick bullets) starting at . The two coordinates of the 60D internal coordinates are shown in a Ramachandran diagram, with adapted axes to the searched reaction patway. The projected gradient paths correspond with the profiles (a), (b), and the lower curve of (c) of Fig. 7. The connected bold points correspond with the CG+ results of Fig. 9. The “better” one (more left and below) is for ; it is the lower curve of Fig. 9. The other chain of points belongs to . The two SPs (*) and the intermediate minimum (+) are included.

Approximation of Newton trajectories for alanine dipeptide between the minima and C5, see text. Methods: projected and inverse gradient (thin points) starting at C5, and modified CG+ optimization (thick bullets) starting at . The two coordinates of the 60D internal coordinates are shown in a Ramachandran diagram, with adapted axes to the searched reaction patway. The projected gradient paths correspond with the profiles (a), (b), and the lower curve of (c) of Fig. 7. The connected bold points correspond with the CG+ results of Fig. 9. The “better” one (more left and below) is for ; it is the lower curve of Fig. 9. The other chain of points belongs to . The two SPs (*) and the intermediate minimum (+) are included.

Approximation of two NTs for alanine dipeptide between the C5 minimum and intermediate minimum, , and from there to the minimum , as well. Method: modified CG+ optimization. Two coordinates are shown in a Ramachandran diagram. Ten nodes are used for every string, and thresholds and 0.003 33, respectively. The inlay shows the energy profile of both paths in the order of their calculation.

Approximation of two NTs for alanine dipeptide between the C5 minimum and intermediate minimum, , and from there to the minimum , as well. Method: modified CG+ optimization. Two coordinates are shown in a Ramachandran diagram. Ten nodes are used for every string, and thresholds and 0.003 33, respectively. The inlay shows the energy profile of both paths in the order of their calculation.

Approximation of four NTs for alanine dipeptide between the C5 minimum and . Method: modified CG+ optimization. The linear combinations of coordinate are used for predictor steps, and five nodes are calculated for every string. The thresholds are , 0.007 5, 0.01, and 0.02 from bottom to top.

Approximation of four NTs for alanine dipeptide between the C5 minimum and . Method: modified CG+ optimization. The linear combinations of coordinate are used for predictor steps, and five nodes are calculated for every string. The thresholds are , 0.007 5, 0.01, and 0.02 from bottom to top.

## Tables

**Scheme 2** SP search in GAUSSIAN03 by Berny’s method.

**Scheme 2** SP search in GAUSSIAN03 by Berny’s method.

**Scheme 3** Shell script for an cluster: Calculation of NTs

**Scheme 3** Shell script for an cluster: Calculation of NTs

**Scheme 4** Head of GAUSSIAN03 input for corrector steps (file *oben*).

**Scheme 4** Head of GAUSSIAN03 input for corrector steps (file *oben*).

**Scheme 5** Shell script for the alanine dipeptide molecule: Calculation of NTs

**Scheme 5** Shell script for the alanine dipeptide molecule: Calculation of NTs

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