Method Introduction

There are various methods that could search local minimums for a function. This locally lowest value search could be called numerical optimization, which could happen in one dimension or  more than one dimension. Normally, the later one is more difficult.

Here, only optimization in one dimension is studied and methods adopted are bisection method and Newton’s method.

These two methods are iterative and can only provide approximations, which is controlled by the convergence criterion. An initial estimate is needed to use these two methods and no guidance is provided by the methods themselves. So the following content will study how the initial guess changes convergence properties of these two methods.

The function used here is f(x)=exp(-x)cos(x). Ranges selected for bisection method are (1.2, 2.4), (1.4, 2.6), (1.6, 2.8), (1.8, 3.0), (2.0, 3.2), (2.2, 3.4). Initial estimates are boundary values of each range for bisection method. Initial estimates selected for Newton’s method are 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4. The initial estimates of Newton’s method are consistent with that of bisection method, which is designed for comparison between these two methods.

A reference local minimum for the studied function is x=2.356194490… . How many decimal places is determined by the precision we set. The convergence criterion used here is 0.000001(1E-06). The range selection for bisection is stopped at (2.2, 3.4) (the change step is 0.2 between each range) otherwise the reference value will fall outside the range.

Another tricky thing is that bisection method has two initial estimates, which form a range for later iteration, while Newton’s method only needs one. So, when we doing comparison, choosing which value for Newton’s method should be paid attention. Since Newton’s method converges to different local minimums with bisection method at 3.0, 3.2, 3.4, smaller boundary values of each range(1.2, 1.4, 1.6, 1.8, 2.0, 2.2) are adopted for Newton’s method when we comparing these two methods at each range.

Calculation Results

The difference of each step ε is defined as x value is this iteration minus x value in last iteration.

Following figures show the calculation results. Full data could be got through the link attach at the end of this post.

Comparison for convergence property between bisection and newton’s method

Range (1.2, 2.4)

Range (1.2, 2.4) is chosen for bisection method, the local minimum is 2.356194.

For New’s method, 1.2 is the initial estimate. The local minimum is  2.356194.

Fig 1 shows the convergence properties of bisection method and Newton’s method. Fig 2 shows a part of Fig1 in order to show the convergence detail.

 Fig1. difference of each step ε vs iteration steps at range (1.2, 2.4)

 

Fig2. part of Fig1 in order to show details of convergence

Both of the methods converge to the same local minimum under the convergence criterion of 1E-06. But the Newton’s method converges needing less steps. The same pattern could be found for following ranges.

Range (1.4, 2.6)

Range (1.4, 2.6) is chosen for bisection method. The local minimum is 2.356194.

For Newton’s method, the initial estimate is 1.4. The local minimum is 2.356194.

Fig 3 and 4 show the convergence properties of the two methods.

Fig 3. difference of each step ε vs iteration steps at range (1.4, 2.6)

 

Fig 4. part of Fig3 in order to show details of convergence

Range (1.6, 2.8)

Range (1.6, 2.8) is chosen for bisection method. The local minimum is 2.356194.

For Newton’s method, the initial estimate is 1.6. The local minimum is 2.356194.

Fig 5 and 6 show the convergence properties of the two methods.

Fig 5. difference of each step ε vs iteration steps at range (1.6, 2.8)

 

Fig 6. part of Fig5 in order to show details of convergence

Range (1.8, 3.0)

Range (1.8, 3.0) is chosen for bisection method. The local minimum is 2.356194.

For Newton’s method, the initial estimate is 1.8. The local minimum is 2.356194.

Fig 7 and 8 show the convergence properties of the two methods.

Fig 7. difference of each step ε vs iteration steps at range (1.8, 3.0)

Fig 8. part of Fig 7 in order to show details of convergence

Range (2.0, 3.2)

Range (2.0, 3.2) is chosen for bisection method. The local minimum is 2.356194.

For Newton’s method, the initial estimate is 2.0. The local minimum is 2.356194.

Fig 9 and 10 show the convergence properties of the two methods.

Fig 9. difference of each step ε vs iteration steps at range (2.0, 3.2)

Fig 10. part of Fig 9 in order to show details of convergence

Range (2.2, 3.4)

Range (2.2, 3.4) is chosen for bisection method. The local minimum is 2.356194.

For Newton’s method, the initial estimate is 2.2. The local minimum is 2.356194.

Fig 11 and 12 show the convergence properties of the two methods.

Fig 11. difference of each step ε vs iteration steps at range (2.2, 3.4)

 

Fig 12. part of Fig 11 in order to show details of convergence

 

Study for convergence property for bisection and Newton’s method, respectively

The Bisection method

Fig 13 shows the convergence property of bisection method at different range. When only one local minimum exits in the ranges, we can say that optimization in different ranges has the same convergence path, which could be confirmed from data in attachment.

Fig 13. difference of each step ε vs iteration steps for bisection method at different ranges

Newton’s method

Besides 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, Newton’s method could get the same local minimum 2.356194 at 2.4, 2.6, 2.8 for the initial estimate.So the new initial guesses are included for the comparison, which is shown in Fig 14. Only around the local minimum, we can say that the convergence is faster if the initial estimate is closer to the local minimum. When the initial estimate is far from the local minimum, the argument is not right. For example, the convergence is faster when the initial estimate is 1.2 compared with the situation when the initial estimate is 2.8, while 2.8 is closer to 2.356194. The explanation could be that the Newton’s method uses the tangent of the first-order derivative function of the original function to find points where  the first-order derivative is zero. This search is determined by the initial estimate and the curve’s shape of the derivative function.

Fig 14. difference of each step ε vs iteration steps for Newton’s method at different initial estimates

 

Another local minimum found by Newton’s method

When the initial estimate is 3.0, 3.2, 3.4, Newton’s method converges to different numbers from 2.356194. Results are shown in Table 13. The convergence value could be a local minimum when the second-order derivative is positive. So when the initial estimate is 3.0, 3.2, 3.4, the convergence values are local maximum actually. Then some negative initial estimates are tried. The new local minimum could be -3.926990.

the initial estimate	convergence value	second-order derivative
other	2.356194	0.134
3.0	-63.617251	-0.613
3.2	11.780972	-1.081
3.4	5.497787	-0.005
-3.0	2.356194	0.134
-4.0	-3.926990	71.776

Table 1 some trials to find other local minimums

 

Conclusion

Comparing how fast (‘fast’ in this post means needing less iteration steps)  the convergence happens for the two methods makes more sense if they converge to the same value. So the initial estimate is chosen to let bisection and Newton’s method converge to the same value. Under this condition, Newton’s method converges faster, in other words, needs less iteration steps than bisection method. A little change for the initial estimate that does not change the convergence value will still give the same fact that Newton’s method converges with less iteration steps.

Bisection method studied respectively, if the ranges chosen for initial estimates has only one minimum, changing the initial estimate will not change the convergence path, which could be seen in Fig 13.

For Newton’s method, if the initial estimate is close to the local minimum, the convergence could need less iteration steps, which however does not mean relatively far initial estimates will guarantee a convergence with more steps. The convergence property is related to the function’s property as well. For the function studied in this post exp(-x)cos(x), we can have other variations like exp(-ax)cos(x) or exp(-x)cos(ax), in which a is a positive number. As we can see, the function has two parts, ‘exp’ and ‘cos’. the first part determines the functions value especially when x takes values far away from 0 and the second part determines the oscillation period. If we introduce parameter ‘a’ into this function, we can expect we will have different function curves and different local minimums.

Here are some plots showing the curves of f(x) = exp(-x)cos(x) and its two variations in different ranges.

Fig 15. f(x) in range(1, 3), we can see the local minimum at 2.356194.

Fig 16. f(x) in range (-5, 5), we can see the local minimum at -3.926990.

Fig 17. f(x) in range (-100, 100), we can see another local minimum near -100.

Fig 18. f(x) and its variations: exp(-2x)cos(x) and exp(-x)cos(2x) in range (-5, 5)

Fig 19. f(x) and its variations: exp(-2x)cos(x) and exp(-x)cos(2x) in range (-100, 100)

From these plots, we can verify above-mentioned local minimums for f(x) and see f(x) and its variations have different curve shapes, which would affect convergence of Newton’s method.

Another point of view is if the initial estimate is far enough, a new extremum might be found. But whether it is a local minimum is determined by the sigh of its second-order derivative.

Reference

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 2002.

Support information

Calculation data could be got through this link:

https://psu.box.com/s/h1ha1f482o6qbj0j3idk4b5nrmbxkhld

 

 

Introduction

The Pt(110) surface has been experimentally shown to undergo surface reconstruction. The mechanism for surface reconstruction is to have missing alternate rows of the top layer of the surface in a \((2\times1)\) surface unit cell. The aim of this study is to verify that the reconstructed Pt(110) is energetically favorable by computing and comparing the surface energy of Pt(110) surface with and without surface reconstruction

All energy calculations were carried out using plane-wave based DFT. The GGA based PBE functional was used to treat the exchange-correlation effects. The ion and core were treated using ultrasoft pseudopotentials generated on the fly (OTFG ultrasoft) with Koelling-Harmon relativistic treatments. Pseudo atomic calculations for Pt treated the 4f14 5s2 5p6 5d9 6s1 electrons as valence electrons and the electrons in lower levels were treated as part of the frozen core.

Calculation of Lattice constant for bulk Pt

Before attempting to compute surface energies we must verify we have the correct lattice parameters for the conventional unit cells used to build the desired surfaces. By correct lattice parameter(s) we refer to the lattice parameter(s) that minimizes the total energy calculated by our DFT code consistent with our choice of exchange correlation functional and pseudo potentials. Using any other lattice constant will introduce errors in our calculations by introducing artificial stress.

Bulk Pt naturally occurs in the FCC crystal structure. The conventional unit cell and primitive unit cell of bulk Pt are shown in figure 1. The first step of our calculations was to geometry optimize the primitive unit cell of bulk Pt, allowing the lattice parameters to relax. For this calculation we used a cutoff of \(480\;eV\) and \(k\) point separation of \(0.04\;\mathring{A}^{-1}\). These calculation parameters were the default parameters for the ‘ultra-fine’ setting for CASTEP calculations as implemented by Materials Studio™. The geometry optimization yielded a lattice constant of \(3.967\;\mathring{A}\) for the conventional unit cell (\(2.805\;\mathring{A}\) for the primitive unit cell).

The next step in our investigation is to test the convergence of the total energy with respect to the cutoff energy and \(k\) point sampling. Figure 2 shows the plots for the results of the calculations to test for convergence. From these plots we see that total energy of the primitive unit cell converges to a tolerance of \(1\;m\,eV\) for cutoff energy of \(480\;eV\) and k point separation of \(0.025\;\mathring{A}^{-1}\).

Calculation of surface energy of Pt(110) surface without surface reconstruction

The surface energy of Pt(110) was calculated by cleaving the optimized conventional unit cell of bulk Pt along the (110) plane. The cleaved unit cell was repeated infinitely in directions parallel to the cleaved plane and a finite number of times perpendicular to the cleaved plane to obtain a slab of desired thickness (in atomic layers). Both surfaces of the slab have Pt(110) surface. Now to employ plane wave based DFT calculations a vacuum slab unit cell was created containing a unit cell of the infinite slab (in directions parallel to the surface) and a vacuum layer in the direction normal to the surface. Figure 3 shows such a vacuum slab unit cell containing a 5 atomic layer thick slab separated by \(10\;\mathring{A}\) of vacuum from the next parallel slab.

Once we have defined a vacuum slab as depicted in figure 3, we fix the position of the bottom two layers to mimic bulk and allow the remaining layers of atoms to relax. In our DFT calculations for the slab unit cell, a self-consistent dipole correction was introduced to correct for the dipole rising from the asymmetry of the cell due to constraining the two bottom layers.

The surface energy (\(\sigma\)) can be calculated using,

\begin{equation} \sigma=\frac{ E_{slab}-n\,E_{bulk} }{A}. \label{surf_energy}\end{equation}

Where, \(E_{slab}\) is the total energy of the vacuum slab unit cell, \(E_{bulk}\) is the total energy of a primitive unit cell of Pt in bulk, \(A\) is the surface area of the slab in the vacuum slab unit cell and \(n\) is the number of Pt atoms in the vacuum slab unit cell.

First we need to determine the convergence tolerance of our calculation of  vacuum slab unit cell energy with respect to cutoff energy and k point sampling used. Here we chose to use the cutoff energy and k point separation used for the bulk unit cell, i.e. \(480\;eV\) and \(0.025\;\mathring{A}^{-1}\) respectively. Table 01 below lists the values obtained for the total energy of the vacuum slab unit cell (6 atomic layer slabs with vacuum separation of \(8\;\mathring{A}\)) for different cutoff and k point sampling around the chosen values. From this table we see that the total energy of the vacuum slab unit cell has a convergence tolerance of \(\sim\;10\;meV\).

Table 01 : Tests for convergence of total energy with respect to k point separation and cutoff energy

Cutoff Energy (eV)	k point separation (1/Å)	Total energy of vacuum slab (eV)
440	0.025	-78303.015
480	0.025	-78303.017
520	0.025	-78303.017
480	0.020	-78303.009
480	0.025	-78303.017
480	0.030	-78303.003

For a given choice of cutoff energy and k point sampling, the thickness of the vacuum layer and the slab thickness will be the two parameters that effect the convergence of the surface energy. In order to test the convergence of the surface energy with respect to these parameters we compute \(\sigma\) while varying vacuum thickness (for a slab with 5 atomic layers) and also  \(\sigma\) while varying slab thickness (with a vacuum thickness of \(10\;\mathring{A}\)). The plots for these calculations are shown in figure 4. From the plots presented in figure 4 we can say for a vacuum thickness of \(8\;\mathring{A}\) and a slab thickness of 6 atomic layers the surface energy converges to (keeping in mind that the convergence tolerance of the total energy of the vacuum slab with respect to energy cutoff and k point sampling is \(10\;m\,eV\)),

\begin{equation}\sigma_{unreconstructed}=125\pm 1 \;\frac{m\,eV}{\mathring{A}^2}.\label{sigma_unrec}\end{equation}

 

Calculation of surface energy of Pt(110) surface with surface reconstruction

The vacuum slab unit cell containing the reconstructed Pt(110) depicted in figure 5 was geometry optimized and the resulting total energy was used to calculate the surface energy of the reconstructed surface using \eqref{surf_energy}. Ideally we would want to retest the convergence tolerance with respect to cutoff energy, k point sampling, slab thickness and vacuum thickness. However given the computational cost of doing these calculations we chose to assume that the convergence tolerance for the surface energy is the same as surface energy calculation for the unreconstructed surface. With this assumption we get,

\begin{equation}\sigma_{reconstructed}=119\pm 1 \;\frac{m\,eV}{\mathring{A}^2}.\label{sigma_rec}\end{equation}

Conclusion

From the results given in equations\eqref{sigma_unrec} and \eqref{sigma_rec} we  see that,

\begin{equation}\sigma_{reconstructed}<\sigma_{unreconstructed}.\label{result}\end{equation}

This indicates that the “missing row” reconstructed surface of Pt(110) is energetically favorable compared to Pt(110) without surface reconstruction. This finding is in line with experimental evidence supporting this claim [1].

References

[1] H. Niehus, “Analysis of the Pt(110)–(1 × 2) surface reconstruction”, Surf. Sci., 145 (1984), pp. 407-418

For this problem we want to find the formation energy of bcc CuPd alloy from fcc Cu and fcc Pd. Then use that to justify why CuPd is the favored low temperature crystal structure of Pd and Cu when they are mixed with this stoichiometry.

Methods

The optimal lattice parameter for fcc Cu, fcc Pd and bcc CuPd was be found, then using the optimal lattice parameter the cohesive energy of each crystal was calculated.

The exchange-correlation functional used was PBE, the pseudopotential used was OTFG ultrasoft and the relativistic treatment was that that of Koelling-Harmon. The pseudo atomic configuration of the pseudo potential was 3d10 4s1 for the copper atom and 4s2 4p6 4d10 5s0.5 for the palladium atom.

Convergence tests with respect to the number of k-points and the energy cutoff energy were preformed for each of the three crystals and the results of which can be found in the following section. To get a convergence of 0.01eV in the cohesive energy a energy cutoff of 4000eV and a k-point grid of 10x10x10 were used. Also, the SCF tolerance was set to 1.0e-6eV/atom with a convergence window of three steps, all other parameters where kept to the quality fine preset for CASTEP.

For the calculation the space group for Cu and Pd was Fm-3m and Pm-3m for the CuPd alloy.

Convergence

k-points

For each crystal the k-point grid was varied and the energy calculated, the results shown in Figures 1,2 and 3.

From these three graphs one can see that a k-point grid of 10x10x10 will converge the cohesive energy of all the crystals to at least 0.01ev

Energy Cutoff

For each crystal the energy cutoff was varied and the energy calculated, the results shown in Figures 4, 5 and 6. The k-point grid was selected to be 11x11x11 for each of the crystals to ensure that the convergence of the k-point grid did not effect the convergence of the energy cutoff.

From these three graphs we can see that an energy cutoff of 5000eV converged the cohesive energies to at least 0.02eV, the limiting case being the CuPd alloy in Figure 6.

Optimization

For each of the three crystals, the lattice parameter was optimized in CASTEP, using the exchange correlation functional PBE, a k-point grid of 10x10x10, a cutoff energy of 5000eV, the BFGS algorithm for geometry optimization with full cell optimization. For the convergence tolerance for the energy was 1e-5 eV per atom, for the max force was 0.03 eV per Å, for the max stress was 0.05 GPa, and for the max displacement was 0.001 Å.

For both the fcc Cu and fcc Pd crystal CASTEP changed the unit cell to reduce the computational cost, in the new cell a, b and c are still equal, but the angles α, β, γ, changed from 90° to 60°.

The results of the optimizations are listed in Table 1.

Table 1: Lattice parameters after optimization of the crystals unit cell

Crystal	Lattice Parameter
fcc Cu	2.566
fcc Pd	2.786
bcc CuPd	3.013

Results

For each of the three crystals the energy of the crystal was taken from the last cycle of the geometry optimization and can be found in Table 2.

Table 2: Cohesive Energies of optimized crystal structures, the Final Free Energy is per unit cell

	Final free energy (E-TS) (eV)	Energy of Free Cu atom (eV)	Energy of Free Pd atom (eV)	Cohesive Energy (eV)
Cu	-1680.93	-1677.14		-3.79
Pd	-3493.26		-3490.46	-2.79
CuPd	-5174.43	-1677.14	-3490.46	-6.83

From the cohesive energies in Table 2, the formation energy of the bcc CuPc alloy was found to be -0.24eV.

From the literature we know that the bcc CuPd alloy is the favored low temperature crystal structure of Pd and Cu when they are mixed with this stoichiometry, this would mean that the CuPd alloy is more thermodynamically  stable than the two separate mono-atomic crystals, leading to a formation energy that is negative. This matches with the results of the calculations.

In this project, the optimal geometry of individual water molecules was found using CASTEP geometry optimization calculations [1]. These calculations utilized the GGA Perdew Burke Ernzerhof (PBE) functional and the Koellig-Harmon atomic solver, which was used to find  psuedopotentials for the hydrogen 1s1 electrons and on the oxygen 1s2 2s2 2p4 electrons. In these calculations, the H atoms had the 1s1 electrons used as valence electrons and the 2s2 2p4 electrons of the O atoms were used as the valence electrons. Additionally, the geometry optimization calculations were performed with a 0.01 eV/Å force tolerance.

Fig. 1. Water Molecule

Determination of Lattice Size

Because CASTEP relies on plane waves, one must construct a lattice to find a molecule’s geometry. Additionally, this lattice must be large enough that the neighboring molecules do not interact in a way that would significantly alter their geometry.  Thus, in performing these calculations, one must ensure that their chosen lattice constant is sufficiently large that the molecule geometry converges past the desired tolerance. In this case, the atomic distances converged to 0.01Å with a 9Å lattice constant, which was used for the primary calculation.

(a)

(b)

Fig. 2. Graph of atomic distances vs. Lattice Constant for (a) OH (b) HH.

Table 1. Effects of lattice constant on bond lengths and angles.

Determination of Energy Cutoff

In order to find the desired energy cutoff for our calculations, the energy cutoff was varied until the distances between atoms converged to 0.01Å. A 435eV cutoff energy was found to be sufficient, and a cutoff of 630eV was used for the main calculation.

(a)

(b)

Fig. 3. Graph of atomic distances vs. cutoff for (a) OH (b) HH.

Table 2. Effects of energy cutoff on atomic distances

Determination of k-space Sampling

To ensure that our k-space sampling was large enough, we checked that the atomic distances converged to 0.01Å at the desired sampling. A sampling of 2x2x2 points, which, after symmetry considerations, resulted in a 4 point total sampling, was found to be sufficient.

(a)

(b)

Fig. 4. Graph of atomic distances vs. cutoff for (a) OH (b) HH.

Table 3. Comparison of total k-space sampling and atomic distances. A single k point was found to be sufficient, and a 4 point (2x2x2) sampling was used in the primary calculation.

Symmetry Considerations

When performing geometry optimization calculations, one must carefully consider the symmetry of the system – in the event that the molecule contains certain symmetries, the forces between atoms may cancel and result in a calculation-breaking local minimum. In this case, when the calculation is performed with the hydrogen atoms perfectly symmetric about an axis (see fig. 5), the calculation fails; the geometry remains symmetric, with OH bond lengths of 0.943Å. However, by starting from slightly modified, symmetry breaking starting positions, we can find values that properly converge.

Fig. 5. A crystal of symmetric H2O molecules with a 9Å lattice spacing. The symmetry about the central oxygen atom causes the calculation to converge to the wrong value.

Conclusion

Utimately, these calculations found that H2O has an OH bond length of 0.97Å with a bond angle of 104.2 degrees. These closely match previously found experimental data  of 0.9572Å and 104.52 degrees [2].

References

In this project, H2O and HCN molecules were optimized starting with different initial structures, in order to explore the effect of initial structure guess on optimization process and final optimized geometry.

Methods
The geometry optimization was performed in Material Studio with CASTEP Calculation Package [1]. The functional of Perdew Burke, and Ernzerhoff was employed [2]. Since H2O and HCN are two dimensional molecules, two degrees of freedom, including bond length and angle need to be considered for optimization.

Determine Lattice Parameter
To mimic a gas phase molecule, a supercell representing empty space needs to be built. Fig. 1a shows the energy change with lattice parameter and Fig. 1b shows the computational time change with lattice parameter. Considering computational cost and accuracy, lattice parameter of 10 Å was chosen for further study, as energy change between 10 Å and 9 Å is only 0.001 eV, and energy change between 10 Å and 12 Å is also 0.001 eV.


Fig 1 a, Energy change with lattice parameter; 1 b, computational time with lattice parameter

Determine Energy Cutoff
Similar to lattice parameter determination, energy cut-off determination was based on computational cost and accuracy, which were shown in Fig 2. Energy cut-off of 750 eV was chosen for further studies as energy tolerance is less than 0.005 eV.


Fig 2 a, Energy change with energy cut-off; 1 b, computational time with energy cut-off

H2O structure
Influence of initial bond length on optimization
In this section, we will explore how initial O-H bond length affect optimization by fixing H-O-H bond angle 110.37 degree and changing H-O bond length.

Fig 3. Initial structure of H2O molecule
We can see from Table 1, the optimized O-H bond length is 0.97 Å. As initial guess molecule geometry getting closer to the optimized molecular geometry, the computational time and number of iteration will decrease. There is no big difference on final energies and optimized H-O-H bond angle for different initial structures.

Influence of initial bond angle on optimization
In this section, we will explore how initial H-O-H bond angle affect optimization by fixing H-O bond 0.97 Å, as shown in Fig 4.

Fig 4. Fixed H-O bond length and changing H-O-H bond angle
From Table 2, we can see the optimized O-H bond length is 0.97 Å and H-O-H bond angle is 104.23. As initial guess molecule geometry getting closer to the optimized molecular geometry, the computational time and number of iteration will decrease. There is no big difference on final energies and optimized O-H bond length for different initial structures.

HCN structure
Influence of initial H-C bond length on optimization
In this section, we will explore how initial C-H bond length affect optimization by fixing C-N bond length of 1.51 Å and H-C-N bond angle of 180, as shown in Fig 5.

Fig 5. Fixed C-N bond length and H-C-N bond angle
We can see from Table 3, the optimized C-H bond length is 1.075 Å and H-C-N bond angle is about 179.9 degree. As initial guess molecule geometry getting closer to the optimized molecular geometry, the computational time and number of iteration will decrease. There is no big difference on final energies and optimized C-N bond length for different initial structures.

Influence of initial H-C-N bond angle on optimization
In this section, we will explore how initial H-C-N bond angle affect optimization by fixing C-N bond length of 1.159 Å.
It is worth to note that in Table 4, the first 4 rows we were attaching H to N in order to make the corresponding angle. It turned out that such initial structure will generate C-N-H molecule after optimization, as shown in Fig 6. From the first 4 rows, we can conclude as initial geometry getting closer to optimized structure, the optimization time and number of iteration will decrease. But for the last row data, we attached H to C atom and H-C-N angle is very close to optimized structure, we found energy of H-C-N molecule is lower than C-N-H molecule.


Fig 6. optimized C-N-H molecule from Table 5 first 4 rows data.

Conclusion
A good estimate of atomic geometry will greatly increase DFT calculation speed.

Reference
[1] “First principle methods using CASTEP” Zeitschrift fuer Kristallographie 220(5-6) pp. 567-570 (2005)
[2] Perdew, J. P; Burke, K; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865-3868